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 $$H = V \rtimes G$$ be the semidirect product corresponding to the representation. Then, $$\sigma$$ extends to an automorphism $$\sigma'$$ of $$H$$ such that $$\sigma'$$ 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 $$G$$. $$H = V \rtimes G$$ is the corresponding semidirect product.

To prove: $$\sigma$$ extends to an automorphism $$\sigma'$$ 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 $$H = V \rtimes G$$ is a finite group containing $$G$$.
 * 2) Since $$\sigma$$ is finite-extensible, it extends to an automorphism $$\sigma'$$ 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, $$\sigma'$$ restricts to an automorphism of $$V$$.