In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the existence of nontrivial automorphisms and nontrivial elementary embeddings of the universe in models of set theory without Foundation take the following form:
{$id_{V}$}$\subseteq$$Aut(V)$$\subseteq$$Eem(V)$
where $id_{V}$ is just the identity mapping from $V$ to $V$, $Aut(V)$ are the automorphisms from $V$ to $V$,and $Eem(V)$ are the elementary embeddings from $V$ to $V$.
In fact, they prove that there are models of $ZFC^{-f}$ that realize each of these four separating refinements of {$id_{V}$}$\subseteq$$Aut(V)$$\subseteq$$Eem(V)$:
i). {$id_{V}$}=$Aut(V)$=$Eem(V)$
ii). {$id_{V}$}$\subsetneq$$Aut(V)$=$Eem(V)$
iii).{$id_{V}$}=$Aut(V)$$\subsetneq$$Eem(V)$
iv). {$id_{V}$}$\subsetneq$$Aut(V)$$\subsetneq$$Eem(V)$
My question is simply this:
Are there models of $NGB+{\lnot}AC$ that realize each of the four separating refinements?