Group property-conditionally quotient-pullbackable automorphism

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.

Related properties

 * Group property-conditionally pushforwardable automorphism
 * Group property-conditionally extensible automorphism