How can i share apk file in my app (send app itself)

Copy the file from the source directory to a new directory. Rename the file while copying and share the copied file. Delete the temp file after share is complete.

 private void shareApplication() {
    ApplicationInfo app = getApplicationContext().getApplicationInfo();
    String filePath = app.sourceDir;

    Intent intent = new Intent(Intent.ACTION_SEND);

    // MIME of .apk is "application/vnd.android.package-archive".
    // but Bluetooth does not accept this. Let's use "*/*" instead.
    intent.setType("*/*");

    // Append file and send Intent
    File originalApk = new File(filePath);

    try {
        //Make new directory in new location
        File tempFile = new File(getExternalCacheDir() + "/ExtractedApk");
        //If directory doesn't exists create new
        if (!tempFile.isDirectory())
            if (!tempFile.mkdirs())
                return;
        //Get application's name and convert to lowercase
        tempFile = new File(tempFile.getPath() + "/" + getString(app.labelRes).replace(" ","").toLowerCase() + ".apk");
        //If file doesn't exists create new
        if (!tempFile.exists()) {
            if (!tempFile.createNewFile()) {
                return;
            }
        }
        //Copy file to new location
        InputStream in = new FileInputStream(originalApk);
        OutputStream out = new FileOutputStream(tempFile);

        byte[] buf = new byte[1024];
        int len;
        while ((len = in.read(buf)) > 0) {
            out.write(buf, 0, len);
        }
        in.close();
        out.close();
        System.out.println("File copied.");
        //Open share dialog
        intent.putExtra(Intent.EXTRA_STREAM, Uri.fromFile(tempFile));
        startActivity(Intent.createChooser(intent, "Share app via"));

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

Update: this method does not work anymore and throws exception if you implement it. Since android N, we should use content providers if we want to have access to files in memory(like the apk file). For more information please visit this Guide. Although the whole idea of copying and renaming and sharing the copied version is still valid.

You can use this function, test on api 22 and 27

    private void shareApplication() {
        ApplicationInfo app = getApplicationContext().getApplicationInfo();
        String filePath = app.sourceDir;

        Intent intent = new Intent(Intent.ACTION_SEND);

        // MIME of .apk is "application/vnd.android.package-archive".
        // but Bluetooth does not accept this. Let's use "*/*" instead.
        intent.setType("*/*");

        // Append file and send Intent
        File originalApk = new File(filePath);

        try {
            //Make new directory in new location=
            File tempFile = new File(getExternalCacheDir() + "/ExtractedApk");
            //If directory doesn't exists create new
            if (!tempFile.isDirectory())
                if (!tempFile.mkdirs())
                    return;
            //Get application's name and convert to lowercase
            tempFile = new File(tempFile.getPath() + "/" + getString(app.labelRes).replace(" ","").toLowerCase() + ".apk");
            //If file doesn't exists create new
            if (!tempFile.exists()) {
                if (!tempFile.createNewFile()) {
                    return;
                }
            }
            //Copy file to new location
            InputStream in = new FileInputStream(originalApk);
            OutputStream out = new FileOutputStream(tempFile);

            byte[] buf = new byte[1024];
            int len;
            while ((len = in.read(buf)) > 0) {
                out.write(buf, 0, len);
            }
            in.close();
            out.close();
            System.out.println("File copied.");
            //Open share dialog
//          intent.putExtra(Intent.EXTRA_STREAM, Uri.fromFile(tempFile));
            Uri photoURI = FileProvider.getUriForFile(this, BuildConfig.APPLICATION_ID + ".provider", tempFile);
//          intent.putExtra(Intent.EXTRA_STREAM, Uri.fromFile(tempFile));
            intent.putExtra(Intent.EXTRA_STREAM, photoURI);
            startActivity(Intent.createChooser(intent, "Share app via"));

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

This only happens because it is saved by base.apk name. To share it as per your need you have to just copy this file into another directory path and rename it over there. Then use new file to share.

This file path[file:///data/app/com.yourapppackagename/base.apk] in data folder is having only read permissions so you can't rename .apk file over there.