  We investigate the braid group representations arising from categories of
representations of twisted quantum doubles of finite groups. For these
categories, we show that the resulting braid group representations always
factor through finite groups, in contrast to the categories associated with
quantum groups at roots of unity. We also show that in the case of p-groups,
the corresponding pure braid group representations factor through a finite
p-group, which answers a question asked of the first author by V. Drinfeld.
