# Are the inner automorphisms the only ones that extend to every overgroup?

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. $\Phi\circ i=i\circ \phi$?

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The answer is that the inner automorphisms are indeed characterized by the property of the existence of extensions to larger groups containing the original group. I learned as much from this blog entry (in Russian). The reference is Schupp, Paul E., "A characterization of inner automorphisms.", Proc. Amer. Math. Soc. 101 (1987), no. 2, 226--228.

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Where by "The answer is "no"", you are referring to the question in the body of the post, rather tha nthe question in the title. –  Theo Johnson-Freyd Dec 4 '09 at 21:42
Oh, so this was ambiguous. I've edited it out. Thanks for pointing this to my attention. –  Leonid Positselski Dec 4 '09 at 22:48