Examples of modules $M$ such that $M^r \cong_R M^s \iff \exists A \in \mathcal A \; r,s \in A$, where $\mathcal A$ is a partition of $\mathbb N$

Question

Let $\mathcal A$ be a partition of $\mathbb N := \{1,2,3,\dots\}$. I would like to know:

Is there an example of a ring $R$ and an $R$-module $M$ such that $M^r$ is isomorphic to $M^s$ as left $R$-modules if and only if $r$ and $s$ are in the same set of $\mathcal A$, i.e. $$M^r \cong_R M^s \iff \exists A \in \mathcal A \; r,s \in A$$

We can find a related question on MO. Examples with $M=R$ or $R = \Bbb Z$ are particularly welcome. This can also be related. This answer provides a very good example.

Thank you very much!