Enable sync for the settings from the Extension Settings API.

chrome/browser/extensions/extension_settings_api.h

Index: chrome/browser/extensions/extension_settings_api.h
diff --git a/chrome/browser/extensions/extension_settings_api.h b/chrome/browser/extensions/extension_settings_api.h
index 6ddfd87170f4db89c9583cf3cec6f71398ad8de9..0058d27b7cd93d81fd360f98daeb170654baaaaa 100644
--- a/chrome/browser/extensions/extension_settings_api.h
+++ b/chrome/browser/extensions/extension_settings_api.h
@@ -10,6 +10,7 @@
 #include "chrome/browser/extensions/extension_function.h"
 #include "chrome/browser/extensions/extension_settings.h"
 #include "chrome/browser/extensions/extension_settings_storage.h"
+#include "chrome/browser/extensions/syncable_extension_settings_storage.h"
 
 // Superclass of all settings functions.
 class SettingsFunction : public AsyncExtensionFunction {
@@ -40,7 +41,7 @@ class SettingsFunction : public AsyncExtensionFunction {
 
  private:
   // Callback method from GetStorage(); delegates to RunWithStorageImpl.
-  void RunWithStorage(ExtensionSettingsStorage* storage);
+  void RunWithStorage(SyncableExtensionSettingsStorage* storage);
 };
 
 class GetSettingsFunction : public SettingsFunction {