  We establish braided tensor equivalences among module categories over the
twisted quantum double of a finite group defined by an extension of a group H
by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also
prove that the canonical ribbon structure of the module category of any twisted
quantum double of a finite group is preserved by braided tensor equivalences.
We give two main applications: first, if G is an extra-special 2-group of width
at least 2, we show that the quantum double of G twisted by a 3-cocycle w is
gauge equivalent to a twisted quantum double of an elementary abelian 2-group
if, and only if, w^2 is trivial; second, we discuss the gauge equivalence
classes of twisted quantum doubles of groups of order 8, and classify the
braided tensor equivalence classes of these quasi-triangular quasi-bialgebras.
It turns out that there are exactly 20 such equivalence classes.
