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

Given a natural number n ≥ 1 and a number field K, we show the existence of an integer l_0 such that for any prime number l ≥ l_0 , there exists a finite extension F/K, unramified in all places above l, together with a principally polarized abelian variety A of dimension n over F such that the resulting l-torsion representation ρ_{A,l} : G_F → GSp(A[l]) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(F_l) as the Galois group of a finite tame extension of number fields F' /F such that F is unramified above l.