P-quotient-pullbackable automorphism

Definition
Let $$P$$ be a group of prime power order, i.e., a finite $$p$$-group for some prime $$p$$. An automorphism $$\sigma$$ of $$P$$ is termed a p-quotient-pullbackable automorphism if, given any finite $$p$$-group $$Q$$ and a surjective homomorphism $$\rho:Q \to P$$, there exists an automorphism $$\sigma'$$ of $$Q$$ that is a pullback of $$\sigma$$; in other words, $$\rho \circ \sigma' = \sigma \circ \rho$$.

Stronger properties

 * Inner automorphism of a finite $$p$$-group.

Weaker properties

 * Cofactorial automorphism: Any $$p$$-quotient-pullbackable automorphism of a finite $$p$$-group is itself a $$p$$-automorphism.

Facts

 * p-quotient-pullbackable automorphism of elementary abelian group is trivial: This is an immediate corollary of Bryant-Kovacs theorem.