Semidirectly extensible implies linearly pushforwardable for representation over prime field

Statement
Suppose $$F$$ is a prime field (i.e., either a field of prime order or the field of rational numbers), and $$G$$ is a group. Suppose $$V$$ is a finite-dimensional vector space over $$F$$, and $$\rho:G \to GL(V)$$ be a linear representation of $$G$$. Let $$H = V \rtimes G$$ with respect to the induced action of $$G$$ on $$V$$.

Suppose, further, that $$\sigma$$ is an automorphism of $$G$$ that can be extended to an automorphism $$\sigma'$$ of $$H$$ such that $$\sigma'$$ also restricts to an automorphism $$\alpha$$ of $$V$$. Then, $$\rho \circ \sigma = c_\alpha \circ \rho$$ where $$c_\alpha$$ is conjugation by $$\alpha$$ in $$GL(V)$$.

Note that we need the field to be a prime field in order that $$GL(V)$$ is equal to the automorphism group of $$V$$ as a group.

Related facts

 * Quotient-pullbackable implies linearly pushforwardable for representation over prime field

Applications

 * Finite-extensible implies class-preserving
 * Hall-semidirectly extensible implies class-preserving
 * Finite solvable-extensible implies class-preserving
 * Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
 * Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving

Facts used

 * 1) uses::Automorphism group equals general linear group for vector space over prime field
 * 2) uses::Automorphism group action lemma: Suppose $$H$$ is a group, and $$N,G \le H$$ are subgroups such that $$G \le N_H(N)$$. Suppose $$\sigma'$$ is an automorphism of $$H$$ such that the restriction of $$\sigma'$$ to $$N$$ gives an automorphism $$\alpha$$ of $$N$$, and such that $$\sigma'$$ also restricts to an automorphism of $$G$$, say $$\sigma$$. Consider the map:

$$\rho: G \to \operatorname{Aut}(N)$$

that sends an element $$g \in G$$ to the automorphism of $$N$$ induced by conjugation by $$g$$ (note that this is an automorphism since $$G \le N_H(N)$$). Then, we have:

$$\rho \circ \sigma = c_\alpha \circ \rho$$

where $$c_\alpha$$ denotes conjugation by $$\alpha$$ in the group $$\operatorname{Aut}(N)$$.

Proof
Given: A group $$G$$, a homomorphism $$\rho:G \to GL(V)$$ for a finite-dimensional vector space $$V$$ over a prime field $$F$$. $$\sigma$$ is an automorphism of $$G$$ that extends to an automorphism $$\sigma'$$ of $$H$$, such that $$\sigma'$$ also restricts to an automorphism $$\alpha$$ of $$V$$.

To prove: $$\rho \circ \sigma = c_\alpha \circ \rho$$.

Proof: Since $$F$$ is a prime field, $$GL(V)$$ is the whole automorphism group of $$V$$ by fact (1) (in general, it is a proper subgroup). Thus, the element $$\alpha$$, which is a group automorphism of $$V$$, is actually in $$GL(V)$$. Thus, fact (2), setting $$G = G, H = H, N = V, \sigma' = \sigma', \alpha = \alpha, \sigma = \sigma$$, gives the desired result.