Generic Galois extensions for SL2(F5) over Q

Let G(n) be a double cover of either the alternating group A(n) or the symmetric group S-n, and let G(n-1) be the corresponding double cover of A(n-1) or Sn-1. For every odd n >= 3 and every field k of characteristic 0, we prove that the following are equivalent: (i) there exists a generic extension for G(n-1) over k, (ii) there exists a generic extension for Gn over k. As a consequence, there exists a generic extension over Q for the group (A) over tilde (5) congruent to SL2(F-5).