Quite commonly, iPhone contacts are composed of contacts from multiple accounts, such as iCloud, Google, and Outlook. When you add a contact to your iPhone, it doesn't get added to all of these accounts. Rather, your iPhone adds it to the account you specify in your iPhone Settings > Contacts > Default Account.
This is where the Contacts Sync app comes to the rescue. With Contacts Sync, you can sync the contacts from any of your other iPhone contact accounts, such as iCloud, with Google.
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, it's also possible for your Google contacts on iPhone to 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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