Hall-semidirectly extensible implies linearly pushforwardable over prime field

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) uses::Automorphism group action lemma