Pitfalls of Syncing Google with Google

Take caution not to use the app to sync Google with itself, as that won't accomplish anything in most cases.  It only makes sense if there are two separate Google accounts involved, or if the Google contacts on your iPhone are out of sync with your Google contacts on the Internet.

Generally, you should consider syncing your Google account with one of your Apple accounts, such as iCloud.

See this article for more info on how to select an account to sync with Google:
https://playaapps.zendesk.com/hc/en-us/articles/226468868-How-To-Select-2-Accounts-to-Sync