Quotient-pullbackable implies linearly pushforwardable for representation over prime field

Statement
Suppose $$\sigma$$ is an automorphism of a group $$G$$. Suppose $$\rho:G \to GL(V)$$ is a finite-dimensional linear representation of $$G$$ over a prime field (i.e., either the field is of prime order or is the field of rational numbers). Consider the corresponding semidirect product $$H = V \rtimes G$$. Suppose, further, that there exists an automorphism $$\sigma'$$ of $$H$$ such that $$\sigma'$$ restricts to an automorphism $$\alpha$$ of $$V$$, and the automorphism induced on $$G$$ as a quotient equals $$\sigma$$. Then, we have:

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

where $$c_\alpha$$ denotes conjugation by $$\alpha$$.

Note that we need the field to be a prime field in order to ensure that any automorphism of the vector space as an abelian group is also a linear automorphism.

Related facts

 * Semidirectly extensible implies linearly pushforwardable for representation over prime field

Applications

 * Finite-quotient-pullbackable implies class-preserving
 * Conjugacy-separable implies every quotient-pullbackable 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 for quotients: Suppose $$A$$ is an fact about::abelian normal subgroup of a group $$H$$, and $$\sigma'$$ is an automorphism of $$H$$ that restricts to an automorphism $$\alpha$$ of $$A$$. Note that this also shows that $$\sigma'$$ descends to an automorphism, say $$\sigma$$, of $$H/A$$.

Since $$A$$ is abelian, we have an action of the quotient group on it by conjugation (see quotient group acts on Abelian normal subgroup), giving a homomorphism:

$$\rho:H/A \to \operatorname{Aut}(A)$$.

The claim is that:

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

where $$c_\alpha$$ denotes conjugation by $$\alpha$$ as an element of $$\operatorname{Aut}(A)$$.

Proof
Given: A group $$G$$, an automorphism $$\sigma$$ of $$G$$, a representation $$\rho:G \to GL(V)$$ for a vector space $$V$$ over a prime field. There is an automorphism $$\sigma'$$ of $$H = V \rtimes G$$ such that $$\sigma'$$ restricts to an automorphism $$\alpha$$ of $$V$$, and it induces the automorphism $$\sigma$$ on the quotient $$G$$.

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 $$H = H, A = V, \sigma' = \sigma', \alpha = \alpha, \sigma = \sigma$$, gives the desired result.