Let $K=PSL_2(q)$ where $q=p^a$ for some odd prime $p$, and let $G$ be a group such that $G/O(G)\cong K$. (Here $O(G)$ is the largest odd-order normal subgroup of $G$.)

I have a homomorphism $\phi: G\to PGL_n(\overline{\mathbb{F}_r})$ whose image is non-solvable. (Here $r$ is a prime distinct from $p$.) I am interested in giving a lower bound for $n$.

In the case where $O(G)$ is trivial, a classical result originating with work of Frobenius gives a sharp lower bound for $n$, namely $\frac12(q-1)$. (Provided $q\neq9$, but let's ignore this exception.)

I'd like to prove that this bound is best possible, i.e.

Q1. For arbitrary $G$ of the given form, prove that $n\geq \frac12(q-1)$.

I reckon I can do this by brute force using Aschbacher's classification of subgroups of $GL_n(q)$ but I'd prefer a more elegant solution. A couple of easy reductions allow me to assume that $G$ acts absolutely irreducibly, that $G$ is center-less, and that $G$ is a minimal non-split extension of $K$, i.e. there does not exist a proper subgroup $H$ such that $H$ is an extension of $K$.

These reductions led me to wonder about a related (but probably much harder) question:

Q2. What are the center-less minimal non-split extensions of $K$?

I know that examples of these things exist where $G$ is not isomorphic to $K$ (see, for instance this MO question, in particular an example mentioned in the answer of @nkrempel). However I cannot find a systematic treatment of these things in the literature.

*Final note: the word `extension' has two meanings in group theory. In the definition I'm using the group $G$ is an example of an extension of $K$.*