Finitarily extensible automorphism

Symbol-free definition
An automorphism of a group is said to be finitarily extensible if it can be extended to an automorphism of any group containing it, in which it has finite index.

When the starting group is finite, this reduces to the notion of finite-extensible automorphism.

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is said to be finitarily extensible if for any $$H$$ containing $$G$$ such that $$[H:G]$$ is finite, there exists an automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

Stronger properties

 * Extensible automorphism
 * Finite-extensible automorphism

Weaker properties

 * Finitarily extensible endomorphism