I'm working through the 2013 edition of Kunen's Set Theory, and I'm having some trouble with Exercise II.9.10, which shows the consistency of $ZF$ without Foundation and the negation of the axiom of choice relative to $ZFC$ with Foundation replaced by the assertion that there exists an infinite set of Quine atoms. Replacing Kunen's notation with what I think is more standard, he defines for any transitive set T $$V_0(T)=T \\ V_{\alpha+1}(T)=\mathcal{P}(V_\alpha) \\ V_\gamma(T) = \bigcup_{\beta<\gamma}V_\beta(T) \text{ for limit $\gamma$}$$
and $V(T)$ is the union of $V_\alpha(T)$ over all ordinals. Then, letting T be any infinite set of Quine atoms, he defines a subclass $M$ of $V(T)$ consisting of all $y$ such that there exists a finite $A\subseteq T$ such that for all permutations $\pi$ of $T$ which fix everything in $A$, if $\hat{\pi}$ is the extension of $\pi$ to an automorphism of $V(T)$, then $\hat{\pi}(y)=y$.
It is apparent to me why $M$ satisfies all the set existence axioms of $ZF$, why all the well-founded sets of $M$ can be well-ordered, and why $T$ is amorphous in $M$ and therefore contradicts the well-ordering theorem, but Kunen also claims that M is transitive. However, it seems to me that any permutation of T should induce a permutation of $V_1(T)$, the full power set of T, and so any automorphism of $V(T)$ should fix $V_1(T)$, so $V_1(T)\in M$, letting $A$ be empty. However, $V_1(T)$ contains many sets which are not in $M$, such as all infinite coinfinite subsets of $T$, so $M$ is not transitive and there is no reason to expect Extensionality to hold.
What am I missing here?