Поскольку Android компилирует HttpClient 4.0.x вместо 3.x, ниже приведен фрагмент кода для вашей справки.

уверен, как отправлять HTTP-заголовки аутентификации.

У меня есть следующий HttpClient для получения запросов, но не знаете, как я могу отправлять запросы?

public class RestClient extends AsyncTask<String, Void, JSONObject> {
        private String convertStreamToString(InputStream is) {
            /*
             * To convert the InputStream to String we use the
             * BufferedReader.readLine() method. We iterate until the
             * BufferedReader return null which means there's no more data to
             * read. Each line will appended to a StringBuilder and returned as
             * String.
             */
            BufferedReader reader = new BufferedReader(new InputStreamReader(is));
            StringBuilder sb = new StringBuilder();

            String line = null;
            try {
                while ((line = reader.readLine()) != null) {
                    sb.append(line + "\n");
                }
            } catch (IOException e) {
                e.printStackTrace();
            } finally {
                try {
                    is.close();
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }

            return sb.toString();
        }

        /*
         * This is a test function which will connects to a given rest service
         * and prints it's response to Android Log with labels "Praeda".
         */
        public JSONObject connect(String url) {
            HttpClient httpclient = new DefaultHttpClient();

            // Prepare a request object
            HttpGet httpget = new HttpGet(url);

            // Execute the request
            HttpResponse response;
            try {
                response = httpclient.execute(httpget);
                // Examine the response status
                Log.i("Praeda", response.getStatusLine().toString());

                // Get hold of the response entity
                HttpEntity entity = response.getEntity();

                if (entity != null) {

                    // A Simple JSON Response Read
                    InputStream instream = entity.getContent();
                    String result = convertStreamToString(instream);

                    // A Simple JSONObject Creation
                    JSONObject json = new JSONObject(result);

                    // Closing the input stream will trigger connection release
                    instream.close();

                    return json;
                }

            } catch (ClientProtocolException e) {
                e.printStackTrace();
            } catch (IOException e) {
                e.printStackTrace();
            } catch (JSONException e) {
                e.printStackTrace();
            }

            return null;
        }

        @Override
        protected JSONObject doInBackground(String... urls) {
            return connect(urls[0]);
        }

        @Override
        protected void onPostExecute(JSONObject json) {

        }
    }

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