Eu tenho uma única atividade (MainActivity) que estende a barra de ação e implementa o NavigationDrawer. Eu configurei o drawertoggle na própria atividade.

Estou criando novos fragmentos a partir de um fragmento que foi criado por outro fragmento, que por sua vez é criado pela MainActivity. (MainActivity -> HomeFragment -> AnotherFragment).

MainActivity.java

public class MainActivity extends ActionBarActivity implements FragmentDrawer.FragmentDrawerListener, TitleSetter, NavigationDrawerEnabler{
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        myToolbar = (Toolbar) findViewById(R.id.toolbar);
        setSupportActionBar(myToolbar);
        getSupportActionBar().setHomeButtonEnabled(true);
        //getSupportActionBar().setDisplayHomeAsUpEnabled(true);
        drawerFragment = (FragmentDrawer)
                getSupportFragmentManager().findFragmentById(R.id.fragment_navigation_drawer);
        setUp(R.id.fragment_navigation_drawer, (DrawerLayout) findViewById(R.id.drawer_layout), myToolbar);
        mDrawerLayout = (DrawerLayout) findViewById(R.id.drawer_layout);
        drawerFragment.setDrawerListener(this);
        // display the first navigation drawer view on app launch
        displayView(0);

    }


    @Override
    public boolean onCreateOptionsMenu(Menu menu) {
        // Inflate the menu; this adds items to the action bar if it is present.
        getMenuInflater().inflate(R.menu.menu_main, menu);
        return super.OncreateOptionsMenu(menu);
    }

    @Override
    public boolean onOptionsItemSelected(MenuItem item) {
        // Handle action bar item clicks here. The action bar will
        // automatically handle clicks on the Home/Up button, so long
        // as you specify a parent activity in AndroidManifest.xml.


        /*  This function is never called even if the Up caret button is pressed!!   */
     if (mDrawerToggle.isDrawerIndicatorEnabled() &&
                mDrawerToggle.onOptionsItemSelected(item))
            return true;
        switch (item.getItemId()) {
            case android.R.id.home:
                /*This case doesn't occur*/
                return true;
            case R.id.action_settings:
                return true;
            default:
                return super.onOptionsItemSelected(item);
        }
    }

    @Override
    public void onDrawerItemSelected(View view, int position) {
        mDrawerLayout.closeDrawer(containerView);
        displayView(position);
    }

    private void displayView(int position) {
        Fragment fragment = null;
        String title = getString(R.string.app_name);
        switch (position) {
            case 0:
                fragment = new HomeFragment();
                title = getString(R.string.title_home);
                break;
            case 1:
                fragment = new SomeFragment();
                title = getString(R.string.title_something);
                break;
            case 2:
                //fragment = new SomeotherFragment();
                //title = getString(R.string.title_somethings);
                break;
            default:
                break;
        }
        if (fragment != null) {
            FragmentManager fragmentManager = getSupportFragmentManager();
            FragmentTransaction fragmentTransaction = fragmentManager.beginTransaction();
            fragmentTransaction.replace(R.id.container_body, fragment);
            fragmentTransaction.commit();
            getSupportActionBar().setTitle(title);
        }
    }


    @Override
    public void setTitle(String title) {
        getSupportActionBar().setTitle(title);
    }


    public void setUp(int fragmentId, DrawerLayout drawerLayout, final Toolbar toolbar) {
        containerView = findViewById(fragmentId);
        mDrawerLayout = drawerLayout;
        mDrawerToggle = new ActionBarDrawerToggle(this, drawerLayout, toolbar, R.string.drawer_open, R.string.drawer_close) {
            @Override
            public void onDrawerOpened(View drawerView) {
                super.onDrawerOpened(drawerView);
                getSupportActionBar().invalidateOptionsMenu();
            }

            @Override
            public void onDrawerClosed(View drawerView) {
                super.onDrawerClosed(drawerView);
                getSupportActionBar().invalidateOptionsMenu();
            }

            @Override
            public void onDrawerSlide(View drawerView, float slideOffset) {
                super.onDrawerSlide(drawerView, slideOffset);
                toolbar.setAlpha(1 - slideOffset / 2);
            }
        };

        mDrawerLayout.setDrawerListener(mDrawerToggle);
        mDrawerLayout.post(new Runnable() {
            @Override
            public void run() {
                mDrawerToggle.syncState();
            }
        });

    }

    @Override
    public void onBackPressed() {
        super.onBackPressed();
        enableNavigationDrawer(true);
    }

    @Override
    public void enableNavigationDrawer(boolean isEnabled) {
        if ( isEnabled ) {
            mDrawerLayout.setDrawerLockMode(DrawerLayout.LOCK_MODE_UNLOCKED);
            getSupportActionBar().setDisplayHomeAsUpEnabled(false);
            mDrawerToggle.setDrawerIndicatorEnabled(true);
            mDrawerToggle.syncState();

        }
        else {
            mDrawerLayout.setDrawerLockMode(DrawerLayout.LOCK_MODE_LOCKED_CLOSED);
            mDrawerToggle.setDrawerIndicatorEnabled(false);
            getSupportActionBar().setDisplayShowHomeEnabled(true);
            getSupportActionBar().setHomeButtonEnabled(true);
            getSupportActionBar().setDisplayHomeAsUpEnabled(true);
            mDrawerToggle.syncState();
        }
    }
}

HomeFragment.java

public class HomeFragment extends Fragment implements View.OnClickListener {
    @Override
    public View onCreateView(LayoutInflater inflater, ViewGroup container,
                             Bundle savedInstanceState) {
        View rootView = inflater.inflate(R.layout.home, container, false);
        // Inflate the layout for this fragment
        fab = (FloatingActionButton) rootView.findViewById(R.id.fab);
        fab.setOnClickListener(this);
        ((TitleSetter) getActivity()).setTitle("Home");
        return rootView;
    }

    @Override
    public void onAttach(Activity activity) {
        super.onAttach(activity);
    }

    @Override
    public void onDetach() {
        super.onDetach();
    }

    @Override
    public void onClick(View v) {

        switch (v.getId()) {
            case R.id.fab:
                ((NavigationDrawerEnabler) getActivity()).enableNavigationDrawer(false);
                Fragment newFragment = new NewWord();
                FragmentManager fragmentManager = getFragmentManager();
                FragmentTransaction fragmentTransaction = fragmentManager.beginTransaction();
                fragmentTransaction.replace(R.id.container_body, newFragment);
                fragmentTransaction.addToBackStack(null);
                // Commit the transaction
                fragmentTransaction.commit();
                break;
        }

    }


}

AnotherFragment.java

public class NewWord extends Fragment {
    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setHasOptionsMenu(true);
    }

    @Override
    public void onCreateOptionsMenu(Menu menu, MenuInflater inflater) {
        super.onCreateOptionsMenu(menu, inflater);
    }

    @Override
    public boolean onOptionsItemSelected(MenuItem item) {
        // Get item selected and deal with it
        /*Not even this*/
        Log.d("KEY: ", String.valueOf(item.getItemId()));
        switch (item.getItemId()) {
            case android.R.id.home:
        /* Doesn't execute*/
                Log.d("Fragment", "I'm here");
                getActivity().onBackPressed();
                return true;
            default:
                return super.onOptionsItemSelected(item);
        }
    }

    @Override
    public View onCreateView(LayoutInflater inflater, ViewGroup container,
                             Bundle savedInstanceState) {
        View rootView = inflater.inflate(R.layout.new_word, container, false);
        return rootView;
    }
    @Override
    public void onAttach(Activity activity) {
        super.onAttach(activity);
        ((TitleSetter) activity).setTitle("Text");
    }

Também o layout da gaveta

<android.support.v4.widget.DrawerLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:id="@+id/drawer_layout"
android:layout_width="match_parent"
android:layout_height="match_parent">


<LinearLayout
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:orientation="vertical">

    <LinearLayout
        android:id="@+id/container_toolbar"
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:orientation="vertical">

        <include
            android:id="@+id/toolbar"
            layout="@layout/toolbar" />
    </LinearLayout>

    <FrameLayout
        android:id="@+id/container_body"
        android:layout_width="fill_parent"
        android:layout_height="0dp"
        android:layout_weight="1" >

    </FrameLayout>


</LinearLayout>
<fragment
    android:id="@+id/fragment_navigation_drawer"
    android:name="com.calgen.wordbook.activity.FragmentDrawer"
    android:layout_width="@dimen/nav_drawer_width"
    android:layout_height="match_parent"
    android:layout_gravity="start"
    app:layout="@layout/fragment_nav_drawer"
    tools:layout="@layout/fragment_nav_drawer" />

E menu_main.xml

<menu xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
tools:context=".MainActivity">

<item
    android:id="@+id/action_search"
    android:title="@string/action_search"
    android:orderInCategory="100"
    android:icon="@drawable/ic_action_search"
    app:showAsAction="ifRoom" />

<item
    android:id="@+id/action_settings"
    android:title="@string/action_settings"
    android:orderInCategory="100"
    app:showAsAction="never" />

Na verdade, eu consultei este tutorial para configurar o fragmento da gaveta de navegação:Seção do Android.

Também referi perguntas semelhantes respondidas aqui no stackoverflow:

1Alternando entre a imagem da Gaveta de Navegação Android e o cursor para cima ao usar fragmentos

2. Meu problema é semelhante a isso:Indicador de exibição da gaveta de navegação do Android para fragmentos de nível inferior. - Não pude comentar aqui, pois não tenho 50 repetições. Eu implementei o que foi dito exatamente nos comentários.

Mas ainda assim, clicar no cursor para cima não executa a função onOptionItemSelected, mesmo que o botão homebutton esteja ativado.!

Agora estou adicionando meu styles.xml personalizado, pode haver algum erro nisso?

<?xml version="1.0" encoding="utf-8"?>
<resources>
<style name="MyMaterialTheme" parent="MyMaterialTheme.Base"></style>

<style name="MyMaterialTheme.Base" parent="Theme.AppCompat">
    <item name="windowNoTitle">true</item>
    <item name="windowActionBar">false</item>
    <item name="android:activatedBackgroundIndicator">@drawable/nav_background</item>
    <item name="colorPrimary">@color/colorPrimary</item>
    <item name="colorPrimaryDark">@color/colorPrimaryDark</item>
    <item name="android:textColorPrimary">@color/textColorPrimary</item>
    <item name="colorAccent">@color/colorAccent</item>
</style>
</resources>

Também o toolbar.xml personalizado

<?xml version="1.0" encoding="utf-8"?>
<android.support.v7.widget.Toolbar
xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:local="http://schemas.android.com/apk/res-auto"
android:id="@+id/toolbar"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:minHeight="?attr/actionBarSize"
android:background="?attr/colorPrimary"
local:theme="@style/ThemeOverlay.AppCompat.Dark.ActionBar"
local:popupTheme="@style/ThemeOverlay.AppCompat.Light" />