  We construct countable groups $G$ with the following new degree of
W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting
a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$,
then the groups $G$ and $\Lambda$ must be virtually isomorphic. Moreover, we
allow both group von Neumann algebras to be twisted by an arbitrary
$2$-cocycle. We also give examples of II$_1$ factors $N$ that are
indecomposable in every sense: they are not virtually isomorphic to any cocycle
twisted groupoid von Neumann algebra.
