My question is related to https://mathoverflow.net/questions/289643/does-every-elementary-embedding-jv-to-v-in-zfa-arise-from-a-self-injection-o
Let $M$ be a transitive model of ZFCA (ZFC with atoms) and suppose that $j: M \rightarrow M$ is an elementary embedding that moves some ordinal in $M$.
Let $A^M$ be the set of atoms in $M$ and let us assume that $A^M$ is NOT a set in $M$ (see Noah's answer if we assume $A^M$ is a set). What do we know about $j$'s action on $A^M$? In particular, is it possible that $j \upharpoonright A^M$ is the identity map?
First, to forestall a reasonable concern readers might have, note that since $M$ is not assumed to contain $V$ as its "pure" part the Kunen inconsistency does not apply. For example, it doesn't prevent elementary embeddings of $L$ into itself. In more technical language, as Asaf Karagila comments above there is no well-founded model of $\mathsf{ZFCA}$ with an elementary self-embedding moving some ordinal which is amenable over that model.
Now on to the actual question:
We have a lot of freedom here. For example, suppose $M$ satisfies "There is a bijection between $A$ and $\omega$." In this case since elementary embeddings can't move $\omega$ we do in fact get $j[A]=A$: consider the action of $j$ on some bijection $f:A\rightarrow\omega$. (This is basically the same argument as that $M\cap \omega_1$ is downwards-closed whenever $M$ is an elementary submodel of $V$.) And this generalizes to any model where (in $M$) the set of atoms is in bijection with some cardinal $<crit(j)$. On the other hand, basically for the same reason if $M\models\vert A\vert>crit(j)$ then $f[A]\subsetneq A$.
So that gives a complete description of what happens if, from the perspective of $M$, the set $A$ of atoms is in bijection with some "pure" set. This of course leaves open the question of what happens if there is no bijection in $M$ between $A$ and any pure set; I'm not sure what things look like in that case.