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 $P(G)$?