By default, the Contacts Sync app syncs your iCloud contacts with Google, even if you have more than just iCloud contacts on your iPhone. If you have multiple contact accounts on your iPhone, and you want to sync all of them with Google, go to Contacts Sync > Accounts > iPhone Account to Sync, and select "All." Then sync again. If you continue to have trouble, send an email to firstname.lastname@example.org
Due to an iOS bug, sometimes your Google contacts on iPhone can fall out of sync with your Google contacts on the web, at https://contacts.google.com/. Thankfully, you can fix that with Contacts Sync. To do so, go to Contacts Sync > Accounts > iPhone Account to Sync, and choose your Google account. Then go to Contacts Sync > Sync > Google (iOS) -> Google (web).
This help article relates to the iPhone app called "Contacts Sync" which can be downloaded here:
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