Finite-extensible implies semidirectly extensible for representation over finite field of coprime characteristic

Statement

Suppose $G$ is a finite group and $\sigma$ is a finite-extensible automorphism of $G$. Suppose $\rho$ is a linear representation of $G$ over a finite field whose characteristic does not divide the order of $G$. Let $V$ be the corresponding vector space and $H=V⋊G$ be the semidirect product corresponding to the representation. Then, $\sigma$ extends to an automorphism ${\sigma }^{\prime }$ of $H$ such that ${\sigma }^{\prime }$ also restricts to an automorphism of $V$.

Proof

Given: A finite group $G$, a finite-extensible automorphism $\sigma$ of $G$, a representation of $G$ over a finite field of characteristic $p$ that does not divide the order of $G$. $H=V⋊G$ is the corresponding semidirect product.

To prove: $\sigma$ extends to an automorphism ${\sigma }^{\prime }$ of $H$ that also restricts to an automorphism of $V$.

Proof:

1. Since $V$ is a finite-dimensional vector space over a finite field, it is finite, and hence $H=V⋊G$ is a finite group containing $G$.
2. Since $\sigma$ is finite-extensible, it extends to an automorphism ${\sigma }^{\prime }$ of $H$.
3. Finally, $V$ is a normal $p$-Sylow subgroup of $H$, and hence, characteristic in $H$ (it can be described as the set of elements whose order divides $p$). Thus, ${\sigma }^{\prime }$ restricts to an automorphism of $V$.