Use task runner in version_updater_win.cc.

chrome/browser/ui/webui/help/version_updater_win.h

 // Copyright 2015 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.
 
 // Windows implementation of version update functionality, used by the WebUI
 // About/Help page.
 
 #ifndef CHROME_BROWSER_UI_WEBUI_HELP_VERSION_UPDATER_WIN_H_
 #define CHROME_BROWSER_UI_WEBUI_HELP_VERSION_UPDATER_WIN_H_
 
@@ skipping 20 matching lines @@
   // UpdateCheckDelegate:
   void OnUpdateCheckComplete(const base::string16& new_version) override;
   void OnUpgradeProgress(int progress,
                          const base::string16& new_version) override;
   void OnUpgradeComplete(const base::string16& new_version) override;
   void OnError(GoogleUpdateErrorCode error_code,
                const base::string16& html_error_message,
                const base::string16& new_version) override;
 
  private:
-  void BeginUpdateCheckOnFileThread(bool install_update_if_possible);
+  void DoBeginUpdateCheck(bool install_update_if_possible);
 
   // A task run on the UI thread with the result of checking for a pending
   // restart.
   void OnPendingRestartCheck(bool is_update_pending_restart);
 
   // The widget owning the UI for the update check.
   gfx::AcceleratedWidget owner_widget_;
 
   // Callback used to communicate update status to the client.
   StatusCallback callback_;
 
   // Used for callbacks.
   base::WeakPtrFactory<VersionUpdaterWin> weak_factory_;
 
   DISALLOW_COPY_AND_ASSIGN(VersionUpdaterWin);
 };
 
 #endif  // CHROME_BROWSER_UI_WEBUI_HELP_VERSION_UPDATER_WIN_H_