If a group automorphism $\phi:G\to G$ is an inner automorphism then it fixes conjugacy classes in the sense that $\phi(C)=C$ for every conjugacy class $C$.
When I encountered this for the first time (long ago) I wondered whether the converse of this might be true. Only since short I am wiser because someone provided me a nice counterexample. It concerned the permutation group on the set of positive integers.
However he could not give me (yet) a finite group as counterexample.
This gave birth to my question:
If $\phi:G\to G$ is an automorphism that fixes conjugacy classes and moreover group $G$ is finite then can we conclude that $\phi$ is an inner automorphism?
Thank you for taking notice and sorry on forehand if it is a duplicate.
It is true for finite simple groups. This is a result of Walter Feit and Gary Seitz (1984). Seitz communicated this to me in a letter, but it was formally published afterwards. I needed the result for classifying finite groups whose complex irreducible characters are all primitive, see here (in particular $(6.1)$).
Article: the result of Feit and Seitz appeared in On finite rational groups and related topics, Illinois Jour. of Math., vol. 33 (1989), pages 103-131. It needs the CFSG.