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



Are the inner automorphisms the only ones that extend to every overgroup?
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](http://mathreader.livejournal.com/22934.html) (in Russian). The reference is *Schupp, Paul E.*, [**A characterization of inner automorphisms**](http://dx.doi.org/10.2307/2045986), Proc. Am. Math. Soc. 101, 226-228 (1987). [ZBL0627.20018](https://zbmath.org/?q=an:0627.20018).. 


