# Extensible local isomorphism

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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.