Symmetric Galois groups under specialization

Given an irreducible bivariate polynomial f (t, x) ∈ ℚ[t, x], what groups H appear as the Galois group of f (t₀, x) for infinitely many t₀ ∈ ℚ? How often does a group H as above appear as the Galois group of f(t₀, x), t₀ ∈ ℚ? We give an answer for f of large x-degree with alternating or symmetric Galois group over ℚ(t). This is done by determining the low genus subcovers of coverings X˜→ℙℂ1 with alternating or symmetric monodromy groups.