Enable sync for the settings from the Extension Settings API.

chrome/chrome_browser.gypi

Index: chrome/chrome_browser.gypi
diff --git a/chrome/chrome_browser.gypi b/chrome/chrome_browser.gypi
index af5f9d84cd8622d706eaaec797cf66f8911074a9..d8f4008b197c82418d7df95dcdd6d45516899f47 100644
--- a/chrome/chrome_browser.gypi
+++ b/chrome/chrome_browser.gypi
@@ -1067,8 +1067,12 @@
         'browser/extensions/extension_proxy_api_helpers.h',
         'browser/extensions/extension_rlz_module.cc',
         'browser/extensions/extension_rlz_module.h',
+        'browser/extensions/extension_setting_sync_data.cc',
+        'browser/extensions/extension_setting_sync_data.h',
         'browser/extensions/extension_settings.cc',
         'browser/extensions/extension_settings.h',
+        'browser/extensions/extension_settings_ui_wrapper.cc',
+        'browser/extensions/extension_settings_ui_wrapper.h',
         'browser/extensions/extension_settings_api.cc',
         'browser/extensions/extension_settings_api.h',
         'browser/extensions/extension_settings_noop_storage.cc',
@@ -1077,6 +1081,8 @@
         'browser/extensions/extension_settings_storage.h',
         'browser/extensions/extension_settings_storage_cache.cc',
         'browser/extensions/extension_settings_storage_cache.h',
+        'browser/extensions/extension_settings_sync_util.cc',
+        'browser/extensions/extension_settings_sync_util.h',
         'browser/extensions/extension_settings_leveldb_storage.cc',
         'browser/extensions/extension_settings_leveldb_storage.h',
         'browser/extensions/extension_service.cc',
@@ -1166,6 +1172,8 @@
         'browser/extensions/pending_extension_manager.h',
         'browser/extensions/sandboxed_extension_unpacker.cc',
         'browser/extensions/sandboxed_extension_unpacker.h',
+        'browser/extensions/syncable_extension_settings_storage.cc',
+        'browser/extensions/syncable_extension_settings_storage.h',
         'browser/extensions/theme_installed_infobar_delegate.cc',
         'browser/extensions/theme_installed_infobar_delegate.h',
         'browser/extensions/user_script_listener.cc',
@@ -2087,6 +2095,8 @@
         'browser/sync/glue/do_optimistic_refresh_task.h',
         'browser/sync/glue/extension_data_type_controller.cc',
         'browser/sync/glue/extension_data_type_controller.h',
+        'browser/sync/glue/extension_setting_data_type_controller.cc',
+        'browser/sync/glue/extension_setting_data_type_controller.h',
         'browser/sync/glue/frontend_data_type_controller.cc',
         'browser/sync/glue/frontend_data_type_controller.h',
         'browser/sync/glue/generic_change_processor.cc',