Let $\phi \in Sym(\mathbb{N})$, where $\mathbb{N}$ is a set of countable atoms which can be represented by natural numbers and $Sym(\mathbb{N})$ is the symmetric group of $\mathbb{N}$. Let $X$ be a set from a von Neumann-like hierarchy built from these atoms instead of the empty set (like in permutation models of ZFA-- this can be found in Jech's book "The Axiom of Choice"). Let $Sym(\mathbb{N})$ act on such sets with the natural action (moving the atoms). If $\phi X \not= X$, does this imply that there exists a transposition $(a b) \in Sym(\mathbb{N})$ such that $(a b) X \not= X$?
In other words, is it the case that if $X$ is changed upon action by a permutation (moving possibly infinite atoms), then we can always find a permutation moving only finitely many atoms which changes it?
No. For instance, let $E\subset\mathbb{N}$ be the even numbers and let $X$ be the set of subsets $A\subset\mathbb{N}$ such that all but finitely many elements of $A$ are contained in $E$. Then $\phi X\neq X$ if $\phi$ is a permutation that swaps the even numbers and the odd numbers. However, $X$ is fixed by any finite-support permutation.