By the time I got into genealogy the last surviving Geo's children (my father & aunt) were already elderly with fading memories plus Geo distinctly disliked speaking about his past. My aunt is gone now and dad is getting senile.
My father always thought his father (Geo) might have changed his name, but Dad could never say exactly where that thought had come from. Also there were stories to the effect that Geo had a step brother, possibly named Carter.
This admittedly is not a great fit but it may just explain some missing links. Perhaps George Frederick GARDENER was born George Frederick COX?