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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?

Robin
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    @user297024 I would consider posting your comment as an answer. I realize it is old hat to you, but the question is somewhat natural. Many people know about the fractional linear transformations forming the group of automorphisms, but the classification of its finite subgroups need not be as widely known. Of course: A) strictly your call, B) somebody here may have described that list of finite groups in a context other than Galois theory of the rational function field. – Jyrki Lahtonen Jan 14 '24 at 16:22
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    Like here Qiaochu Yuan describes the classification, but doesn't say much about it. – Jyrki Lahtonen Jan 14 '24 at 16:29

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