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The following problem appears in the study of braided autoequivalences of representation categories of Drinfeld doubles of finite groups.

Let $G$ be a finite group. An outer automorphism $\alpha$ of $G$ is called class preserving if $\alpha$ induces a trivial permutation of the set of conjugacy classes of $G$. Examples of groups having such automorphisms were first constructed by Burnside in 1913 and many more were constructed since then, see, e.g., a survey
http://www.personal.psu.edu/msm344/blogs/psu/research/burnside-involve-FINAL.pdf by Brooksbank and Mizuhara.

I am interested in automorphisms with a more restrictive property. Namely, consider the set $P(G)$ of classes of pairs of commuting elements of $G$ under simultaneous conjugation. That is, $P(G)$ consists of sets of the form $ \{(xax^{-1},\,xbx^{-1})\mid x\in G \}$, where $a,b\in G,\, ab=ba$.

Question: Is there an example of a finite group $G$ and an outer automorphism $\alpha$ of $G$ such that $\alpha$ induces a trivial permutation of $P(G)$?

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  • $\begingroup$ Because I don't know if I understand completely the question, can you tell me how the following fails: $G=A_5$ and $\alpha$ conjugation by the transposition $(1,2)$. I only ask because commuting elements in $G$ are of the same (prime) order. $\endgroup$
    – Steve D
    Commented Sep 24, 2013 at 3:32
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    $\begingroup$ @SteveD: The elements of order $5$ split into two conjugacy classes in $A_5$, and these are exchanged by $(1,2)$. In particular, $a$ and $a^2$ are conjugate in $S_5$ but not in $A_5$ for $a$ of order $5$. Thus $(1,2)$ moves the sets of the form $\{(xax^{-1},1)\mid x\in G\}$ or $\{(xax^{-1},xa^2x^{-1})\mid x\in G\}$. $\endgroup$ Commented Sep 24, 2013 at 10:33

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