Extraspecial and critical implies whole group

Statement
Suppose $$G$$ is a fact about::group of prime power order, and $$H$$ is a fact about::critical subgroup of $$G$$ that is also an fact about::extraspecial group. Then, $$H = G$$.

Critical subgroup
A characteristic subgroup $$H$$ of a finite $$p$$-group $$G$$ is termed critical if it satisfies the following conditions:


 * 1) $$\Phi(H) \le Z(H)$$, viz the Frattini subgroup is contained inside the center (i.e., $$H$$ is a Frattini-in-center group).
 * 2) $$[G,H] \le Z(H)$$ (i.e., $$H$$ is a commutator-in-center subgroup of $$G$$).
 * 3) $$C_G(H)= Z(H)$$ (i.e., $$H$$ is a self-centralizing subgroup of $$G$$).

Facts used

 * 1) Extraspecial commutator-in-center subgroup is central factor

Proof
Given: A finite $$p$$-group $$G$$, a critical subgroup $$H$$ that is also extraspecial.

To prove: $$H = G$$.

Proof: By point (2) of the definition of critical subgroup, $$H$$ is a commutator-in-center subgroup of $$G$$. Combining this with fact (1) yields that $$H$$ is a central factor of $$G$$. Thus, $$HC_G(H) = G$$.

Point (3) of the definition of critical subgroup says that $$C_G(H) = Z(H)$$, so $$HZ(H) = G$$, so $$H = G$$, completing the proof.