  This article extends the main results of the publication arXiv:2001.01312 to
the case of a twisted groupoid. More precisely, it gives a decomposition of the
C*-algebra of a twisted locally compact groupoid with Haar system in presence
of a normal subgroupoid. When the normal subgroupoid and the twist over it are
abelian, one obtains another twisted groupoid C*-algebra. This is applied to
C*-algebraic deformation quantization and to multiplier representations of
locally compact abelian groups.
