Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other finite simple group of Lie type in characteristic $p$). Moreover, $V$ is a faithful irreducible $H$-module (equivalently, $V$ is a minimal normal subgroup of $G$ and $C_G(V)=V$).
Ideally, I would like to show that $G$ must be the semidirect product $V\rtimes H$, or $|V|$ must be somewhat large compared to $|H|$. A quick example: when $p=2$ and $H=\operatorname{PSL}(n,2)$, so that $V$ is the natural $n$-dimemsional $\mathbb{F}_2$-module for $H$, according to "U. Dempwolff, On the second cohomology of $\operatorname{GL}(n,2)$", $G$ must be the mentioned semidirect product.
Any suggestions and references are appreciated.