Symmetric Galois Groups Under Specialization

Given an irreducible bivariate polynomial $f(t,x)\in \mathbb{Q}[t,x]$, what groups $H$ appear as the Galois group of $f(t_0,x)$ for infinitely many $t_0\in $\mathbb{Q}$$?Howoftendoesagroup$H$asaboveappearastheGaloisgroupof$f(t_0,x)$,$t_0\in $\mathbb{Q}$$?Wegiveananswerfor$f$oflarge$x$-degree with alternating or symmetric Galois group over $\mathbb{Q}(t)$. This is done by determining the low genus subcovers of coverings $\tilde{X}\rightarrow $\mathbb{P}$^1_{$\mathbb{C}$}$ with alternating or symmetric monodromy groups.