Question:Inner automorphism extending to bigger group

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This question is about inner automorphism | See more questions about inner automorphism

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. Further information: extensible equals inner (specific proof), extensible automorphisms problem (more discussion)