How can I access my GMail contact groups from my iOS 7 device?
As far as I know the (becoming deprecated) exchange-based sync that gmail used never did it.  And the newer carddav sync doesn't support it "yet". I believe the spec they wrote for the protocol does provide the means for contact groups, but google simply has not implemented everything yet. 

Google  does provide such features in their APIs which is a long way of saying you can find a few third-party solutions at the AppStore for the time-being that will sync your gmail groups bi-directionally.  Not sure if there are any free ones and haven't tried any to recommend, unfortunately.