Hall-semidirectly extensible implies linearly pushforwardable over prime field

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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