Group property-conditionally quotient-pullbackable automorphism

This term is related to: extensible automorphisms problem
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Definition

Suppose $\alpha$ is a group property and $G$ is a group satisfying $\alpha$. An automorphism $\sigma$ of $G$ is termed quotient-pullbackable with respect to $\alpha$, or quotient-pullbackable conditional to $\alpha$, if for any group $K$ and surjective homomorphism $\rho:K \to G$ such that $K$ satisfies property $\alpha$, there is an automorphism $\sigma'$ of $K$ such that $\rho \circ \sigma' = \sigma \circ \rho$.

For more information on the best known results and characterization, refer extensible automorphisms problem.

When the groups satisfying $\alpha$ form a subvariety of the variety of groups, this is equivalent to the notion of variety-quotient-pullbackable automorphism for that subvariety.

Also note that any inner automorphism is conditionally quotient-pullbackable with respect to any group property.