Extensible local isomorphism

Definition
Suppose $$G$$ is a group, $$A$$ and $$B$$ are subgroups of $$G$$, and $$\sigma:A \to B$$ is an isomorphism of groups. We say that $$\sigma$$ is an extensible local isomorphism if, for any group $$K$$ containing $$G$$, there exists an automorphism $$\alpha$$ of $$K$$ such that the restriction of $$\alpha$$ to $$A$$ equals $$\sigma$$.

The extensible local isomorphisms conjecture states that an isomorphism of subgroups of $$G$$ is an extensible local isomorphism if and only if it can be extended to an inner automorphism of $$G$$. This is a strong version of the extensible automorphisms conjecture.