Suppose a group $G$ acts faithfully on a set $X$, or equivalently, $G$ is a subgroup of ${\rm Sym}(X)$.

By functoriality, $G$ acts on $P(X), P(P(X)), P(P(P(X))),$ etc. ($P(\cdot)$ means powerset.) Henceforth, I'll omit parentheses.

One can recover $G$ from ${\rm Fix}_G(PPPX)$ because, for example, one can encode a well-ordering of $X$ as an element of $PPX$.

Generally, one cannot recover $G$ from ${\rm Fix}_G(PPX)$. For example, the alternating group and symmetric groups on a finite set will give the same set of fixed elements.

Write $G\sim H$ if both groups act on $X$ and ${\rm Fix}_G(PPX) = {\rm Fix}_H(PPX)$. Equivalently, $$\forall u,v \subset X (\exists g\in G, gu=v \leftrightarrow \exists h\in H, hu=v) \ .$$

**Questions**

When does $G\sim H$ imply $G=H$?

Are there nontrivial examples of $G\sim H$ for infinite $X$? for $X$ of any infinite cardinality?

Is there a classification of such pairs for finite $X$?

Does this phenomenon have a name?

Can one always recover $G$ from ${\rm Fix}_G(PPPX)$ in *ZF*?