Group property-conditionally pushforwardable automorphism

Definition
Suppose $$\alpha$$ is a group property and $$G$$ is a group satisfying $$\alpha$$. An automorphism $$\sigma$$ of $$G$$ is termed pushforwardable with respect to $$\alpha$$, or pushforwardable conditional to $$\alpha$$, if for any group $$H$$ and homomorphism $$\rho:G \to H$$ such that $$H$$ satisfies property $$\alpha$$, there is an automorphism $$\sigma'$$ of $$H$$ 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-pushforwardable automorphism for that subvariety.

Also note that any inner automorphism is conditionally pushforwardable with respect to any group property.

Weaker properties

 * Stronger than::Group property-conditionally extensible automorphism

Related properties

 * Group property-conditionally quotient-pullbackable automorphism