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Here's the version of the question I'm particularly interested in (though I'm also interested in other variants described near the end).

Let $K$ be the function field of a smooth projective variety over $\mathbb{C}$. Let $\pi : X\rightarrow E$ be a $G$-Galois branched cover of an elliptic curve $E$ over $K$, only branched over the origin $O\in E$. Then the residue fields of $\pi^{-1}(O)$ are all isomorphic, and each is Galois over $K$. What are the possible Galois groups that can appear?

Here's what I know: If $\overline{z}$ is a geometric point landing in $\pi^{-1}(O)$ and $G_{\overline{z}}$ denotes its stabilizer, then moreover the residue field of any point in $\pi^{-1}(O)$ is Galois over $K$ with Galois group isomorphic to a subgroup of $C_G(G_{\overline{z}})/G_{\overline{z}}$ (here $C_G$ denotes the centralizer; this follows e.g. from the answer to this question).

I am interested in knowing if we can further restrict the Galois groups that can appear. If $G$ is abelian, then by "twisting" $\pi$, one can obtain any subgroup of $G$ as the residue field of a point in $\pi^{-1}(O)$ for some $G$-Galois cover $\pi$. In this case since $\pi$ must be unramified, $G_{\overline{z}}$ is trivial, and $C_G(G_{\overline{z}})/G_{\overline{z}} = G$, so in this case we cannot further restrict the Galois groups that can appear.

My question involves what happens at the other end of the abelian-ness spectrum:

1. Let $G$ be a nonabelian finite simple group - Is it true that for any subgroup $H \le C_G(G_{\overline{z}})/G_{\overline{z}}$, there exists a field $K$ and a cover $\pi$ as above such that $H$ is the Galois group of a residue field of $\pi^{-1}(O)$?

I've realized that I have no idea how to go about constructing examples where we can bound these Galois groups from below (at least when $K$ contains all relevant roots of unity). Here's a simpler question:

2. Can we find a nonabelian finite simple group $G$, a field $K$ and a $G$-Galois cover $\pi$ as above such that the residue fields of $\pi^{-1}(O)$ are nontrivial extensions of $K$?

I'd also be interested in answers to variants of the above questions where $E/K$ is replaced with a smooth projective variety over $\mathbb{C}$, where instead of asking about the residue fields, I'd be asking about the Galois groups of components of the reduced ramification divisor (still restricting to nonabelian simple $G$).

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