The action of a group $G$ on a set $X$ is called oligomorphic if the diagonal action on $X^n$ has finitely many orbits for each $n$.

Question: Is there an infinite (maybe even finitely generated) group $G$ such that the conjugation action of $G$ on itself is oligomorphic?

The action of a group $G$ on a set $X$ is called oligomorphic if the diagonal action on $X^n$ has finitely many orbits for each $n$.

Question: Is there an infinite (maybe even finitely generated) group $G$ such that the conjugation action of $G$ on itself is oligomorphic?

Edit: In view of the comments, I would already be happy to see an example of an infinite group $G$ whose conjugation action on $G^2$ or $G^3$ has finitely many orbits.