Hall-semidirectly extensible implies linearly pushforwardable over prime field

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., Hall-semidirectly extensible automorphism) must also satisfy the second automorphism property (i.e., linearly pushforwardable automorphism)
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Statement

Suppose $G$ is a finite group and $\sigma$ is a Hall-semidirectly extensible automorphism of $G$. Then, if $p$ is a prime not dividing the order of $G$, $\sigma$ is a linearly pushforwardable automorphism with respect to the prime field of $p$ elements.

Facts used

1. Automorphism group action lemma