Finite-characteristic-semidirectly extensible implies linearly pushforwardable over prime field

Statement
Suppose $$G$$ is a finite group and $$p$$ is a prime not dividing the order of $$G$$. Let $$k$$ be the prime field with $$p$$ elements. Then, any finite-characteristic-semidirectly extensible automorphism of $$G$$ is linearly pushforwardable over $$k$$.

Finite-characteristic-semidirectly extensible automorphism
An automorphism $$\sigma$$ of $$G$$ is termed finite-characteristic-semidirectly extensible if, for any homomorphism $$\rho:G \to \operatorname{Aut}(N)$$ where $$N$$ is a finite group, such that $$N$$ is a characteristic subgroup in the semidirect product $$M = N \rtimes G$$, the automorphism $$\sigma$$ extends to an automorphism of $$M$$.

Linearly pushforwardable automorphism
An automorphism $$\sigma$$ is termed linearly pushforwardable for a group $$G$$ over a field $$k$$ if, for any finite-dimensional linear representation $$\rho:G \to GL(V)$$, there exists $$a \in GL(V)$$ such that for every $$g \in G$$, we have:

$$\rho(\sigma(g)) = a\rho(g)a^{-1}$$

Facts used

 * Automorphism group action lemma