The folloing is an Edit of the previous question.

By stratified [acyclic ZF][1], it is meant acylic ZF but with restricting Separation and Replacement to stratified instances of them and forbid the use of symbol $\mathcal R$ 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? 


  [1]: https://mathoverflow.net/questions/383773/is-acyclic-zf-consistent