  We introduce a generalization of the Dijkgraaf-Witten invariants for cusped
or compact oriented 3-manifolds. We show that the generalized DW invariants
distinguish some pairs of cusped hyperbolic 3-manifolds with the same
hyperbolic volumes and with the same Turaev-Viro invariants. We also present an
example of a pair of cusped hyperbolic 3-manifolds with the same hyperbolic
volumes and with the same homology groups, whereas with distinct generalized DW
invariants.
