Finite-quotient-pullbackable automorphism

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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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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 $\rho \circ \sigma '=\sigma \circ \rho$.

Relation with other properties

Weaker properties

• Class-preserving automorphism: For full proof, refer: Finite-quotient-pullbackable implies class-preserving