TITLE: On non-split extensions of $\mathrm{SL}_d(q)$
QUESTION [7 upvotes]: Throughout this question, the following notation holds: Let $q$ be a power of a prime $p$, and let $d>4$ be a positive integer. Let $G$ be a finite group with a normal subgroup $E$ which is an elementary-abelian $p$-group, and such that $G/E\cong\mathrm{SL}_d(q)$. 
I am interested in the situation where this extension is non-split, i.e. when there is no subgroup $H<G$ such that $H\cong \mathrm{SL}_d(q)$ and $G=EH$.

Q1 Where can I read about such non-split extensions? What are the main results about such groups?

Given that this is rather general, let me propose a more specific version:

Q2 What can one say if one assumes that $|E|=q^d$?

Note: There are a number of results in the literature for a situation similar to this more specific situation. The difference is that people usually consider what happens when the quotient $G/E$ is isomorphic to $\mathrm{GL}_d(q)$, rather than $\mathrm{SL}_d(q)$. Some relevant results are listed in this MO question - note especially that the answer of @ndkrempel suggests that the situation I am considering allows for significantly different behaviour.
Finally, a question about how such groups might embed into larger linear groups. 

Q3. Suppose that $G$ is isomorphic to a subgroup of $\mathrm{SL}_n(q)$. Is it true that if $d>\frac34n$, then the extension is split?

Note: I chose the number $\frac34$ out of thin air. I'd be happy to hear of a `yes' answer, with that $\frac34$ replaced by your favourite positive real number.

REPLY [2 votes]: Here are some further comments, to supplement Derek's focused answer to Q2 as well as my own somewhat cryptic comments.
1) Q2 is fairly narrow, so the kind of ad hoc treatment Derek suggests is quite appropriate.    Note here that you do need to have some information about the $p$-modular representations, for which the dimension bound is crucial.   In general, even for finite special linear groups not much is known about the details of their modular representations.     
2) I share Derek's view that Q1 is too general to admit a sensible answer.   However, there has been a lot of organized study of extensions and cohomology for finite groups of Lie type, including subtle connections with the corresponding ideas for ambient algebraic groups.    Here the 1977 Invent. Math. paper on rational and generic cohomology for which I gave a link is basic.  (This paper combines independent work by CPS and by van der Kallen, whom I had the privilege to introduce to each other soon afterward at an Oberwolfach meeting.) For a much more recent treatment of issues related to $H^2$ along with a useful reference list, see the paper (later published in J. Algebra) here.     The earliest work tends to rely just on finite group techniques, but by now it's natural to think also about algebraic groups and to deal with general groups of Lie type.  There are of course some special features of your groups which might make things easier, but it's a good idea to have the general picture in mind.   (For the algebraic groups, the possible non-existence of Levi factors will be one related theme.)