Question:Inner automorphism extending to bigger group

Q: '''It is clear that an inner automorphism can be extended to any bigger group as an inner automorphism of that group -- simply use the same conjugating element. Are there other automorphisms that can always be extended?'''

A: No, inner automorphisms are the only ones that can be extended. This is hard to prove. The direction you mentioned is that inner implies extensible. The reverse direction -- that any extensible automorphism is inner -- is nontrivial.