Working in $\sf ZF + GC + j:V \xrightarrow {auto} V + \exists \alpha: V_{j(\alpha)} \subsetneq V_\alpha \land \alpha \text{ is limit }$

Of course $j$ is external in the sense that it is not used in instances of separation nor replacement.

Now suppose that $S \subsetneq V_\alpha$ and such that the ordinal rank of $S$ is $\alpha$, and such that $S$ is transitive; then by automorphism of $j$ we have $j(S) \subsetneq V_{j(\alpha)}$, but:

is it always the case that $j(S) \subsetneq S$?

if the answer is to the negative then what are the properties that a proper subset of $V_\alpha$ of rank $\alpha$ should fulfill in order for it to be moved by $j$ to a proper subset of it?

Working in $$\sf ZF + GC + j:V \xrightarrow {auto} V + \exists \alpha: V_{j(\alpha)} \subsetneq V_\alpha \land \alpha \text{ is limit }$$

Of course $j$ is external in the sense that it is not used in instances of separation nor replacement.

Now suppose that $S \subsetneq V_\alpha$ and such that the ordinal rank of $S$ is $\alpha$, and such that $S$ is transitive; then by automorphism of $j$ we have $j(S) \subsetneq V_{j(\alpha)}$, but:

is it always the case that $j(S) \subsetneq S$?

if the answer is to the negative then what are the properties that a transitive proper subset of $V_\alpha$ of rank $\alpha$ should fulfill in order for it to be moved by $j$ to a proper subset of itself?