Modular quasi-Hopf algebras and groups with one involution

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 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.(c) 2022 Elsevier B.V. All rights reserved.