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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". It can be extended to the class of all finite groups using a theorem of Hartley and Robinson, "On finite complete groups". For how to assemble the proof, see the wiki page https://groupprops.subwiki.org/wiki/Finite-extensible_implies_inner