On $p$-groups with automorphism groups related to the exceptional Chevalley groups

Let $\widehat{G}$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\widehat{G}$ over its field of definition. We explore the overgroup structure of $\widehat{G}$ in $\mathrm{G}\mathrm{L}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $\widehat{G}$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $\mathrm{A}\mathrm{u}\mathrm{t}(P)$ on $P\mathrm{/}\mathrm{\Phi }(P)$ is either $\widehat{G}$ or its normaliser in $\mathrm{G}\mathrm{L}(V)$.