informally speaking, what algebraic structure does the set of conjugacy classes of a group carry?

Formally, I'm interested in **natural operations** on conjugacy classes. Let $\mathsf{Grp}$ be the category of groups and $\mathsf{CC} : \mathsf{Grp} \to \mathsf{Set}$ the functor mapping every group to its set of conjugacy classes. Then a natural operation of arity $n \in \mathbb{N}$ is a natural transformations of the form
$$\mathsf{CC}^{\times n} \longrightarrow \mathsf{CC}.$$

Here are the ones that I know of (hat tip to YCor):

- Projecting onto one of the factors.
- In arity $n = 1$, taking a power $x \mapsto x^k$ for fixed $k \in \mathbb{Z}$, which descends to conjugacy classes.
- Composing the first kind with the second.
- In arity $n = 0$, returning the class of the neutral element.

Are there any other ones? In particular, are there nontrivial operations of arity $> 1$?

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