Abelian varieties over number fields, tame ramification and big Galois image

Given a natural number n and a number field K, we show the existence of an integer ell_0 such that for any prime number ellgeq ell_0, there exists a finite extension F/K, unramified in all places above ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting ell-torsion representation rho_{A,ell} from G_F to GSp(A[ell](bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(mathbb{F}_ell) as the Galois group of a finite tame extension of number fields F/F such that F is unramified above ell.