No way to help without photos. That high serial number might be during the "assemble from on hand parts" era after the sale to Ithaca. That made grade identification even more difficult than usual.

In late guns, an H usually has no scene engraving while a G would usually have a dog. Both would usually have pull off fore ends. Either could be ejector.

If it has a rib extension rather than a dolls head, it is more likely to be a DS/I than an H. Many of the later made 20s seem to be of that grade.