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Let $G$ be a finite group whose center $Z(G)$ is trivial. Suppose that the group $\text{Out}(G)$ of outer automorphisms is nontrivial.

Question: Does there always exist an $f \in \text{Aut}(G)$ and a $g \in G$ such that $g$ and $f(g)$ do not lie in the same conjugacy class?

This question can be reformulated in the following way. The automorphism group $\text{Aut}(G)$ acts on the set of conjugacy classes of $G$. As the inner automorphism group $\text{Inn}(G)$ acts trivially on this set, the action descends to the quotient $\text{Out}(G)$. With the same assumptions as above, the question then becomes:

Question: Is the action of $\text{Out}(G)$ on the set of conjugacy classes always nontrivial?

Remark: I restrict myself to the case that $G$ is centerless because this case is of main importance to me. I am not sure though whether there is an easy answer or counterexample to the question in the case that we do not assume $G$ to be centerless.

What I've got so far: the question is positive for the symmetric group $S_{n}$ on $n$ letters for all $n \geq 3$. Indeed, only for $n = 6$ there exists an outer automorphism and this automorphism interchanges the conjugacy class of a cycle of cycle type $(ab)$ and the conjugacy class of a cycle of cycle type $(cd)(ef)(gh)$. Here, $a \neq b$ and the $c, d, e, f, g$ and $h$ are distinct.

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This question is first asked by Burnside;

Such automorphisms are called as class preserving automorphisms and denoted by $A_c$ or $Aut_c(G)$. As you also notice $Inn(G) \leq A_c\leq Aut(G)$. For some groups two of such groups can be equal. Check this link. If you search as "Class preserving automorphisms", you can find many papers related the this topic.

Edit:

In this paper,

In $1947$, G.E.Wall[30] constructed examples ... Interestingly his examples contain $2$ group having class preserving outer automorphisms.

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  • $\begingroup$ I couldn't immediately locate any examples with $A_c = {\rm Aut}(G)$, which is what the question was asking for. They are mainly concerned with examples in which $A_c$ is bigger than ${\rm Inn}(G)$. Also, most of these examples are $p$-groups, which have nontrivial centres, although a paper by Hertweck describes some Frobenius groups with $A_c \ne {\rm Inn}(G)$. $\endgroup$ – Derek Holt Feb 10 '15 at 20:04
  • $\begingroup$ @DerekHolt: In this paper, it is mentioned that G.E.Wall found such two example.(In introduction part,third paragraph) researchgate.net/publication/… $\endgroup$ – mesel Feb 10 '15 at 22:01
  • $\begingroup$ @DerekHolt: But I could not find the mentioned paper. $\endgroup$ – mesel Feb 10 '15 at 22:07
  • $\begingroup$ Try arxiv.org/abs/1002.1359 It says that Heineken found $p$-groups with $A = {\rm Aut}_c(G)$. But I have still found no references to groups with $Z(G)=1$ and $A = {\rm Aut}_c(G)$. $\endgroup$ – Derek Holt Feb 11 '15 at 8:59
  • $\begingroup$ @mesel: Thank you for making me aware of these kind of automorphisms. The article from G.E. Wall that you mention can be found here. $\endgroup$ – Adam Battelle Feb 11 '15 at 12:40

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