Finite-quotient-pullbackable automorphism

Definition
Let $$G$$ be a finite group and $$\sigma$$ be an automorphism of $$G$$. We say that $$\sigma$$ is a finite-quotient-pullbackable automorphism of $$G$$ if, for any surjective homomorphism $$\rho:K \to G$$, there exists an automorphism $$\sigma'$$ of $$K$$ such that $$\rho \circ \sigma' = \sigma \circ \rho$$.

Weaker properties

 * Stronger than::Class-preserving automorphism: