Unfortunately, you will probably have better luck identifying which neighborhood cat left that last set of muddy paw prints on you freshly washed and waxed car!
The letter G may be a clue as to the door manufacturer, but it also may be the hinge mfr stamp. There are dozens, if not hundreds of door manufacturers in this country of both entire door assemblies, and just the doors and sidelights. Many doors have been hung in frames made by the local carpenter or door and cabinet shop. If the door slab is a common style, it may well have been made by a mass manufacturer who sells them to dozens of outfits who put there own glass in them, or hang them in their own frames, and sell them under their own name. If you can access the outside surface of the door jamb, which is hidden by the inner and outer trim, you might find a label.
But whatever your problem is with the door, your best bet is just go talk to a door shop and they can probably fix it without needing to know who made it.