Extensible local isomorphism
Suppose is a group, and are subgroups of , and is an isomorphism of groups. We say that is an extensible local isomorphism if, for any group containing , there exists an automorphism of such that the restriction of to equals .
The extensible local isomorphisms conjecture states that an isomorphism of subgroups of is an extensible local isomorphism if and only if it can be extended to an inner automorphism of . This is a strong version of the extensible automorphisms conjecture.