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?