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

This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |

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 $k$.

Definitions used

Finite-characteristic-semidirectly extensible automorphism

Further information: 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

Further information: 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:

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