If any installed app has push messaging permission, then background launch Chrome

chrome/browser/background/background_application_list_model.cc

 // Copyright (c) 2012 The Chromium Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
 #include "chrome/browser/background/background_application_list_model.h"
 
 #include <algorithm>
 #include <set>
 
 #include "base/stl_util.h"
@@ skipping 259 matching lines @@
 
 // static
 bool BackgroundApplicationListModel::IsBackgroundApp(
     const Extension& extension, Profile* profile) {
   // An extension is a "background app" if it has the "background API"
   // permission, and meets one of the following criteria:
   // 1) It is an extension (not a hosted app).
   // 2) It is a hosted app, and has a background contents registered or in the
   //    manifest.
 
-  // Not a background app if we don't have the background permission.
-  if (!extension.HasAPIPermission(APIPermission::kBackground))
+  // Not a background app if we don't have the background permission or
+  // the push messaging permission
+  if (!extension.HasAPIPermission(APIPermission::kBackground) &&
+      !extension.HasAPIPermission(APIPermission::kPushMessaging) )
     return false;
 
   // Extensions and packaged apps with background permission are always treated
   // as background apps.
   if (!extension.is_hosted_app())
     return true;
 
   // Hosted apps with manifest-provided background pages are background apps.
   if (extension.has_background_page())
     return true;
@@ skipping 115 matching lines @@
          (*old_cursor)->name() == (*new_cursor)->name() &&
          (*old_cursor)->id() == (*new_cursor)->id()) {
     ++old_cursor;
     ++new_cursor;
   }
   if (old_cursor != extensions_.end() || new_cursor != extensions.end()) {
     extensions_ = extensions;
     FOR_EACH_OBSERVER(Observer, observers_, OnApplicationListChanged(profile_));
   }
 }