TITLE: An equivalence relation on group actions
QUESTION [18 upvotes]: 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?

REPLY [2 votes]: The group of pwop (piecewise order-preserving) permutations of $\mathbf{N}$ is equivalent to the full symmetric group (i.e., has the same orbits on the power set $2^\mathbf{N}$. These are permutations for which there's a finite partition such that on every component, the permutation is order-preserving. It is transitive on moieties (infinite subsets with infinite complement) and obviously is also transitive on finite subsets of given cardinality. You can google "transitive on moieties" to find more.