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.

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.