# Possible Galois groups of residue fields of ramification points of a Galois branched cover of curves

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 $$G$$).