Would automorphisms cause nested subset-hood?

Question

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?

Answer 1

Regarding the first question, no:

Let $\beta=j(\alpha)$ and $\beta+\gamma=\alpha$. Define a version of the cumulative hierarchy with the empty set replaced with $\beta$. (That is, let $V_0’=\beta$, $V_1’=\{\beta\}$, and then $V_{\xi+1}’=\mathcal{P}(V_\xi’)\cup\{\beta\}\backslash\{\emptyset\}$ for $\xi>0$, and take unions at limits.) Let $S=V’_{\gamma}$. This gives a counterexample.

Don’t know about the second question...