I've known plenty of girls that age who are able to fire shotguns. Have you ever fired a shotgun? They have very little kick. That's why even kids can shoot them.
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I've shot a 20 gauge shotgun and a 12 gauge shotgun. The 20 gauge didn't have much kick at all, and I could imagine an 11 year old girl being able to shoot one. It's generally what's recommended for younger kids in skeet tournaments.
However, the 12 gauge had a pretty wicked kick, and I know how to hold and fire a gun. Perhaps it's possible for an 11 year old girl to shoot one, but I'd really have to see it to believe it.