Finite-extensible automorphisms of finite groups are inner. The proof is rather non-trivial. For the classes of finite nilpotent and finite soluble groups, this is proved in a paper of Pettet, "[On inner automorphisms of finite groups](https://www.ams.org/journals/proc/1989-106-01/S0002-9939-1989-0968625-8/S0002-9939-1989-0968625-8.pdf)". It can be extended to the class of all finite groups using a theorem of Hartley and Robinson, "[On finite complete groups](https://link.springer.com/article/10.1007/BF01235320)". For how to assemble the proof, see the wiki page https://groupprops.subwiki.org/wiki/Finite-extensible_implies_inner