Group property-conditionally extensible automorphism

Definition
Suppose $$\alpha$$ is a group property and $$G$$ is a group satisfying $$\alpha$$. An automorphism $$\sigma$$ of $$G$$ is termed extensible with respect to $$\alpha$$, or extensible conditional to $$\alpha$$, if for any group $$H$$ containing $$G$$ such that $$H$$ satisfies property $$\alpha$$, there is an automorphism $$\sigma'$$ of $$H$$ whose restriction to $$G$$ equals $$\sigma$$.

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-extensible automorphism for that subvariety.

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

Stronger properties

 * Weaker than::Group property-conditionally pushforwardable automorphism

Related properties

 * Group property-conditionally quotient-pullbackable automorphism

Other related notions

 * Variety-extensible automorphism
 * Own variety-extensible automorphism
 * Extensibility operator
 * Qualified extensibility operator