  This article extends the works of Gon\c{c}alves, Guaschi, Ocampo [GGO] and
Marin [MAR2] on finite subgroups of the quotients of generalized braid groups
by the derived subgroup of their pure braid group. We get explicit criteria for
subgroups of the (complex) reflection group to lift to subgroups of this
quotient. In the specific case of the classical braid group, this enables us to
describe all its finite subgroups : we show that every odd-order finite group
can be embedded in it, when the number of strands goes to infinity. We also
determine a complete list of the irreducible reflection groups for which this
quotient is a Bieberbach group.
