Female stag beetle??
They don't have horns.
Female stag beetles are usually smaller than the males, with smaller mandibles.
and seem about the right size:
Some species grow up to over 12 cm (4.7 in), but most are about 5 cm
There are several species and they vary in size too. Pretty much all beetles can, and do, fly.
Another option would be the Blaps lethifera: