As long as you have iCloud contacts enabled on all of your Apple devices, you only need to run the Contacts Sync app on just one of your devices, to keep iCloud and Google synced on all devices.
To verify that you have iCloud enabled on all of your iOS devices, go to your iPhone/iPad Settings > Contacts > Accounts > iCloud > iCloud > Contacts. On Mac, go to your Mac System Preferences > iCloud > Contacts.
In addition, using this app to sync iCloud and Google from multiple Apple devices can cause minor inconsistencies.
This help article relates to the iPhone app called "Contacts Sync" which can be downloaded here:
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