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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.

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    $\begingroup$ I think the second cohomology groups of all of the classical groups on their natural modules are known. There are some infinite families of non-split extensions, such as the orthogonal groups in chararcteristic $2$, and I think ${\rm SL}(2,2^n)$ for $n>2$. $\endgroup$
    – Derek Holt
    Sep 25, 2022 at 19:23

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I think this can often fail even for the non-trivial representation $V$ of smallest possible dimension.

For a reductive group $G$ over $\mathbf{Z}_p$, the group $G(\mathbf{Z}/p^2 \mathbf{Z})$ will typically be a non-split extension of $G(\mathbf{F}_p)$ by the adjoint representation $V$, and for various $G$ the adjoint representation $V$ will be the smallest non-trivial representation. For example, take $G = E_8$.

An even simpler example is the group $\Gamma = \mathrm{PSL}_2(\mathbf{F}_p)$ for $p > 3$. Then the smallest non-trivial representation over $\mathbf{F}_p$ will be the adjoint representation $V$ of dimension $3$, but $\mathrm{PSL}_2(\mathbf{Z}/p^2 \mathbf{Z})$ gives a non-split extension of $\Gamma$ by $V$, since the lift of any unipotent element in $\Gamma$ will have order $p^2$.

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