  In a previous paper the authors constructed a class of quasi-Hopf algebras
$D^{\omega}(G, A)$ associated to a finite group $G$, generalizing the twisted
quantum double construction. We gave necessary and sufficient conditions,
cohomological in nature, that the corresponding module category
$Rep(D^{\omega}(G, A))$ is a modular tensor category.\ In the present paper we
verify the cohomological conditions for the class of groups $G$ which
\emph{contain a unique involution}, and in this way we obtain an explicit
construction of a new class of modular quasi-Hopf algebras.\ We develop the
basic theory for general finite groups $G$, and also a parallel theory
concerned with the question of when $Rep(D^{\omega}(G, A))$ is super-modular
rather than modular. We give some explicit examples involving binary polyhedral
groups and some sporadic simple groups.
