About automorphism over non-well-founded models of $\sf ZF+V= L(A)$? Answer

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About automorphism over non-well-founded models of $\sf ZF+V= L(A)$?

Can we have a set $A$ and a non-well founded model $M$ of $\sf ZF + V=L(A)$ such that there exists an external automorphism $j$ over $M$ and a non-standard limit ordinal $\alpha \in M$ other $j(L_\alpha(A)) =L_\beta(A)$ other $|\alpha|=|\beta|^+$ ?

Where $L(A)$ is the relatively constructible hierarchy raised over set $A$

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