Let $M$ a non-well founded model of Finite $\sf ZF$, which is $\sf ZF$ with axiom of infinity replaced by the axiom stating that all sets are finite. So there must be a set $\zeta$ that $M$ thinks it's a natural yet it possess's an infinite descending membership chain as seen from the outside of $M$, i.e. we have $\{ \zeta-1, \zeta-2, \zeta-3, ...\} \subset \zeta $, where each $\zeta -n = \zeta -n-1 \cup \{\zeta-n-1\} $ for all $n \in \mathbb Z$.

Since $M$ is a model of $\sf ZF$, so there must be stages: $$....,V_{\zeta-2}, V_{\zeta-1}, V_\zeta, V_{\zeta + 1}, V_{\zeta+2},...$$

Now we know that some models of $\sf ZF$ can have external downward rank-shifting automorphisms $j$ on them, i.e. $j(V_\alpha) =V_\beta$ where $\beta < \alpha$ for some non-standard ordinal $\alpha$.

Can there be a non-well founded model $M$ of Finite $\sf ZF$, that admits an external automorphism $j$ such that: $j(V_{\zeta+n})= V_{\zeta +n -1}$ ?