Invariant special subgroup lemma

Statement
Suppose $$P$$ is a finite $$p$$-group, i.e., a group of prime power order. Suppose $$A$$ is a subgroup of $$\operatorname{Aut}(P)$$ such that $$p$$ does not divide the order of $$A$$. Suppose $$\varphi$$ is a non-identity element of $$A$$. Then, there exists a subgroup $$Q$$ of $$P$$ that is either special or elementary Abelian, with the following properties:


 * $$\varphi$$ acts nontrivially on $$Q/\Phi(Q)$$.
 * $$A$$ acts irreducible on $$Q/\Phi(Q)$$.
 * $$\varphi$$ acts trivially on $$\Phi(Q)$$.

Facts used

 * 1) uses::Structure lemma for p-group with coprime automorphism group having automorphism trivial on invariant subgroups

Proof
Let $$Q$$ be minimal among the nontrivial $$A$$-invariant subgroups of $$P$$ on which $$\varphi$$ acts nontrivially. (Note that this collection of subgroups is nonempty, since $$\varphi$$ acts nontrivially on $$P$$). Fact (1) then yields that $$Q$$ satisfies the required conditions.

Textbook references

 * , Page 183, Theorem 3.8, Section 5.3 ($$p'$$-automorphisms of $$p$$-groups)