  In this paper we discuss generalizations of discrete torsion to noninvertible
symmetries in 2d QFTs. One point of this paper is to explain that there are two
complementary generalizations. Both generalizations are counted by
$H^2(G,U(1))$ when one specializes to ordinary finite groups $G$. However, the
counting is different for more general fusion categories. Furthermore, only one
generalizes the picture of discrete torsion as differences in choices of gauge
actions on B fields. Explaining this in detail, how one of the generalizations
of discrete torsion to noninvertible cases encodes actions on B fields, is the
other point of this paper. In particular, this generalizes old results in
ordinary orbifolds that discrete torsion is a choice of group action on the B
field. We also explain how this same generalization of discrete torsion gives
rise to physically-sensible twists on gaugeable algebras and fiber functors.
