Own-variety-extensible automorphism

Definition
An automorphism $$\sigma$$ of a group $$G$$ is termed an own-variety-extensible automorphism if, for any group $$H$$ containing $$G$$ such that $$H$$ is in the subvariety generated by $$G$$ in the variety of groups, $$\sigma$$ extends to an automorphism of $$H$$.

In other words, an own-variety-extensible automorphism is an automorphism of a group that satisfies the following equivalent conditions:


 * It is variety-extensible for some subvariety of the variety of groups.
 * It is variety-extensible for the subvariety of the variety of groups generated by that particular group.

Stronger properties

 * Weaker than::Inner automorphism
 * Weaker than::Extensible automorphism