Enable sync for the settings from the Extension Settings API.

chrome/browser/extensions/extension_settings_api.cc

 // Copyright (c) 2011 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 "base/bind.h"
 #include "base/values.h"
 #include "chrome/browser/extensions/extension_service.h"
 #include "chrome/browser/extensions/extension_settings_api.h"
 #include "chrome/browser/profiles/profile.h"
 #include "content/browser/browser_thread.h"
 
 namespace {
 const char* kUnsupportedArgumentType = "Unsupported argument type";
 }  // namespace
 
 // SettingsFunction
 
 bool SettingsFunction::RunImpl() {
-  BrowserThread::PostTask(
-      BrowserThread::FILE,
-      FROM_HERE,
-      base::Bind(&SettingsFunction::RunOnFileThread, this));
+  profile()->GetExtensionService()->extension_settings()->RunWithSettings(
+      base::Bind(&SettingsFunction::RunWithSettingsOnFileThread, this));
   return true;
 }
 
-void SettingsFunction::RunOnFileThread() {
+void SettingsFunction::RunWithSettingsOnFileThread(
+    ExtensionSettings* settings) {
   DCHECK(BrowserThread::CurrentlyOn(BrowserThread::FILE));
-  ExtensionSettings* settings =
-      profile()->GetExtensionService()->extension_settings();
   bool success = RunWithStorage(settings->GetStorage(extension_id()));
   BrowserThread::PostTask(
       BrowserThread::UI,
       FROM_HERE,
       base::Bind(&SettingsFunction::SendResponse, this, success));
 }
 
 bool SettingsFunction::UseResult(
     const ExtensionSettingsStorage::Result& storage_result) {
   if (storage_result.HasError()) {
@@ skipping 57 matching lines @@
     AddAllStringValues(*as_list, &string_list);
     return UseResult(storage->Remove(string_list));
   }
   return UseResult(ExtensionSettingsStorage::Result(kUnsupportedArgumentType));
 }
 
 bool ClearSettingsFunction::RunWithStorage(
     ExtensionSettingsStorage* storage) {
   return UseResult(storage->Clear());
 }