The folloing is an Edit of the previous question.

By stratified acyclic ZF, it is meant acylic ZF but with restricting Separation and Replacement to stratified instances of them.

Is Stratified acyclic ZF consistent with having an internal non-trivial automorphism $j$ from $V$ to $V$?

That is we add a new primitive one place function symbol $j$ to the language of ZF, and add the axioms of $j$ being an automorphism over the whole universe of discourse, that is:

Bijectivity: $\forall x \exists y : j(x)=y \\\forall y \exists x: j(x)=y$

Isomorphism: $\forall x \forall y :x \in y \iff j(x) \in j(y)$

Collapse: $\exists \alpha: V_\alpha \subsetneq V_{j(\alpha)}$

Where $``j"$ is allowed to appear in instances of the stratified separation and replacement.

If this is allowed what would be the consistency strength of such addition?