TITLE: Are the inner automorphisms the only ones that extend to every overgroup?
QUESTION [39 upvotes]: Let $H$ be a group. Can we find an automorphism $\phi :H\rightarrow H$ which is not an inner automorphism, so that given any inclusion of groups $i:H\rightarrow G$ there is an automorphism $\Phi: G\rightarrow G$ that extends $\phi$, i.e. $\Phi\circ i=i\circ \phi$?

REPLY [17 votes]: 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 the class of all groups. Martin Pettet later solved it for the class of finite groups, and his proof works for the classes of $p$-groups, finite $p$-groups, finite $\pi$-groups, solvable groups, etc.

Pettet, Martin R., On inner automorphisms of finite groups, Proc. Am. Math. Soc. 106, No. 1, 87-90 (1989). ZBL0675.20015.
Pettet, Martin R., Characterizing inner automorphisms of groups, Arch. Math. 55, No. 5, 422-428 (1990). ZBL0683.20025.

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.