Is the following known to be consistent with some extension of $\text{ZF}$?

There is a transitive model $M$ of $\text{ZF}$ such that there is an external non-trivial elementary embedding $j$ from $P(M)$ to $P(M)$ (where $P$ is the known Power operator) such that $M \subset range(j)$ and such that every definable subset $\kappa$ of $M$ (parameter free or from parameters in $M$) in the language of set theory (i.e. doesn't use the symbol $j$) is sent by $j$ to an element of $M$[i.e. $j(\kappa) \in M$] and such that the symbol $j$ can be used freely in the instances of Replacement and Separation.

To clarify what is meant by "$\kappa $ is definable subset of $M$ from parameters $w_1,..,w_n \in M$", is to mean that: $$ \forall y (y \in \kappa \leftrightarrow y \in M \wedge \phi(y,w_1,..,w_n))$$ for some formula $\phi(y,w_1,..,w_n)$ that is written in the language of set theory.

In particular: is the above consistent with $\text{ZF + Reinhardt cardinal}$?

EDIT: it appears that my original question has some confusion between auto-morphisms and elementary embeddings as it is obvious from the answer below, therefore I'll clarify here what I exactly want.

Can we have the following?

A transitive model $M$ of ZF-Regularity such that we have an external injective function $j$ from $P(M)$ to $P(M)$ such that $j$ is an ismomorphism on $\in$ between $P(M)$ and $range(j)$. Now we demand that $M$ is a subset of $range(j)$, and that every definable subset of $M$ [from parameters in $M$] would be sent by $j$ to an element of $M$.

Where "$\kappa $ is a definable subset of $M$ from parameters $w_1,..,w_n \in M$", is defined as: $$ \forall y (y \in \kappa \leftrightarrow y \in M \wedge \phi(y,w_1,..,w_n))$$ for some formula $\phi(y,w_1,..,w_n)$ that is written in the language of set theory [with the only exception of allowing $j $ to be appear on parameters, so $j(w_i), j^{-1}(w_i)$ are allowed].

We know that $range(j) \neq M$, because the parity of $M$ is different from $P(M)$.

The question is if there is a clear inconsistency with that?

Is the following known to be consistent with some extension of $\text{ZF}$?

There is a transitive model $M$ of $\text{ZF}$ such that there is an external non-trivial elementary embedding $j$ from $P(M)$ to $P(M)$ (where $P$ is the known Power operator) such that $M \subset range(j)$ and such that every definable subset $\kappa$ of $M$ (parameter free or from parameters in $M$) in the language of set theory (i.e. doesn't use the symbol $j$) is sent by $j$ to an element of $M$[i.e. $j(\kappa) \in M$] and such that the symbol $j$ can be used freely in the instances of Replacement and Separation.

To clarify what is meant by "$\kappa $ is definable subset of $M$ from parameters $w_1,..,w_n \in M$", is to mean that: $$ \forall y (y \in \kappa \leftrightarrow y \in M \wedge \phi(y,w_1,..,w_n))$$ for some formula $\phi(y,w_1,..,w_n)$ that is written in the language of set theory.

In particular: is the above consistent with $\text{ZF + Reinhardt cardinal}$?

Is the following known to be consistent with some extension of $\text{ZF}$?

There is a model $M$ of $\text{ZF}$ such that there is an external non-trivial elementary embedding $j$ from $P(M)$ to $P(M)$ (where $P$ is the known Power operator) such that $M \subset range(j)$ and such that every definable subset $\kappa$ of $M$ (parameter free or from parameters in $M$) in the language of set theory (i.e. doesn't use the symbol $j$) is sent by $j$ to an element of $M$[i.e. $j(\kappa) \in M$] and such that the symbol $j$ can be used freely in the instances of Replacement and Separation.

To clarify what is meant by "$\kappa $ is definable subset of $M$ from parameters $w_1,..,w_n \in M$", is to mean that: $$ \forall y (y \in \kappa \leftrightarrow y \in M \wedge \phi(y,w_1,..,w_n))$$ for some formula $\phi(y,w_1,..,w_n)$ that is written in the language of set theory.

In particular: is the above consistent with $\text{ZF + Reinhardt cardinal}$?

Is the following known to be consistent with some extension of $\text{ZF}$?

There is a transitive model $M$ of $\text{ZF}$ such that there is an external non-trivial elementary embedding $j$ from $P(M)$ to $P(M)$ (where $P$ is the known Power operator) such that $M \subset range(j)$ and such that every definable subset $\kappa$ of $M$ (parameter free or from parameters in $M$) in the language of set theory (i.e. doesn't use the symbol $j$) is sent by $j$ to an element of $M$[i.e. $j(\kappa) \in M$] and such that the symbol $j$ can be used freely in the instances of Replacement and Separation.

To clarify what is meant by "$\kappa $ is definable subset of $M$ from parameters $w_1,..,w_n \in M$", is to mean that: $$ \forall y (y \in \kappa \leftrightarrow y \in M \wedge \phi(y,w_1,..,w_n))$$ for some formula $\phi(y,w_1,..,w_n)$ that is written in the language of set theory.

In particular: is the above consistent with $\text{ZF + Reinhardt cardinal}$?