I'm trying to find examples of groups $G$ where there is a non-Brauer pairs have an interesting conjugation action on it. I am currently trying the symmetric group in characteristic 2. The center of $kG$, where $k$ has characteristic 2, will be generated by conjugacy class sums.
I believe it's necessary that the elements of the conjugacy class be 2-regular. For $S_3$ and $S_4$, it's sufficient, but those are pretty small cases. Is there anything we can say in general about when a class sum is an idempotent?
Another way to think about it may be that given a conjugacy class $X$,
$\left(\sum\limits_{x\in X} x\right)^2 = \sum\limits_{x\in X}x$ if and only if $\sum\limits_{x \neq y} xy = 0$, which in characteristic 2, is equivalent to an involution on the set $\{ (x,y) | x \neq y \}$.
If you have other examples of groups whose representations have an interesting Brauer pair structure, I'd be interested in those as well.