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 $t_0\in \mathbb{Q}$? How often does a group $H$ as above appear as the Galois group of $f(t_0,x)$, $t_0\in \mathbb{Q}$? We give an answer for $f$ of large $x$-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.