What is known about the answer to the following question?
What finite groups can be realised as the Galois group of finite Galois extensions of the form:
$$\mathbb{Q}(X)/\mathbb{Q}(h(X))$$
for $h(X)$ a rational function over $\mathbb{Q}$?
E.g. it turns out $C_2 \cong \mathbb{Q}(X)/\mathbb{Q}(X^2)$, $S_3 \cong \mathbb{Q}(X)/\mathbb{Q}\left(\frac{(X^2-X+1)^3}{X^2(X-1)^2}\right)$.
This question has a notable difference to the inverse Galois problem: in the inverse Galois problem we are trying to find the top field $K$ such that $\operatorname{Gal}(K/\mathbb{Q}) \cong G$, in this problem we are trying to find the bottom field $K$ such that $\operatorname{Gal}(\mathbb{Q}(X)/K) \cong G$.
So, what is known of this problem? E.g. in the inverse Galois problem, it is known that any solvable group can arise as a Galois group. Are any properties of groups (e.g. solvable) known to satisfy this? In the inverse Galois problem it is conjectured all finite groups arise. Is there a "popular" conjecture for this question, if it is indeed not known?