If she likes the AR platform (look), you might want to investigate the Ruger SR-22 rifle ( Ruger® SR-22® * Autoloading Rifle Models ). It's a standard 10/22 wrapped up in an AR blanket. This rifle weighs a ton and the factory Ruger sights are crap, but the 10/22 is a well vetted platform. If she ever gets tired of this look, the sky's the limit on making it look like something else.
If your not bird hunting shot guns suck. .22LR is extinct. If you do happen to find some it isn't worth buying.