gr.group theory - Are the inner automorphisms the only ones that extend to every overgroup?

I struggled with the same question for quite some time and solved it for finite groups, only to then discover that it had already been solved. Schupp solved it for arbitrary groups. Martin Pettet later solved it for finite groups, and his proof works for p-groups, finite p-groups, finite pi-groups, solvable groups, etc.

1. On inner automorphisms of finite groups by Martin R. Pettet, Proceedings of the American Mathematical Society, Volume 106,Number 1, Page 87 - 90(May 1989)




2. Characterizing inner automorphisms of groups by Martin R. Pettet, Archiv der Mathematik, Volume 55,Number 5, Page 422 - 428(Year 1990)

These proofs also show that analogous statements are true if we replace injective embeddings with quotient maps (i.e., the only automorphisms that can be pulled back over all quotient maps are the inner ones).

I also have some notes on this and similar problems here: extensible automorphisms problem.