Extensible automorphisms problem

This article describes an open problem in the following area of/related to group theory: [[{{{1}}}]]

Statement

The Extensible automorphisms problem is as follows: given a group $G$, give a characterization of which automorphisms of $G$ are extensible. In other words, describe the group of extensible automorphisms of $G$.

A priori, determining whether or not an automorphism of a group is extensible does not seem to be easy, because it requires quantification over all bigger groups containing the group. Thus, solving the extensible automorphisms problem amounts to removing the every group containing it quantifier to some more manageable quantifier(s).

Associated conjecture

A conjecture associated with the solution to this open problem is: Extensible automorphisms conjecture

The Extensible automorphisms conjecture states that every extensible automorphism of a group is inner, or equivalently, that the automorphism property of being extensible equals the automorphism property of being inner.