Structure of coefficients of polynomials giving a specified Galois group

Question

For any $a = [a_0; \dots; a_n]\in \mathbb{P}^n(\mathbb{Q})$, the corresponding Galois group $G_a$ of $f(X) = a_n X^n + \cdots + a_1 X + a_0\in \mathbb{Q}[X]$ is a subgroup of $S_n$. For a given group $G \subset S_n$, what exactly can be said about the size of the set of points $a$ with $G_a = G$? For $G = S_n$, the answer from here is that $G = S_n$ iff the resolvent of $f$ is irreducible, and this happens for $a$ in a Zariski-dense set by Hilbert's irreducibility theorem. Generalizing from that, is there a specific sense in which $A(X) = \{a\in \mathbb{P}^n(\mathbb{Q}):\, G_a\subset X\}$ has $A(H)\subset A(G)$ "small" in $A(G)$ for $H < G$? That is, is there a result invoking some notion like the dimension of a variety (although we're necessarily working over $\mathbb{Q}$ here), a thin set in the sense of Serre, etc. that makes this vague idea of "smallness" more precise?