  We study Doi-Hopf data and Doi-Hopf modules for Hopf group-coalgebras. We
introduce modules graded by a discrete Doi-Hopf datum; to a Doi-Hopf datum over
a Hopf group coalgebra, we associate an algebra graded by the underlying
discrete Doi-Hopf datum, using a smash product type construction. The category
of Doi-Hopf modules is then isomorphic to the category of graded modules over
this algebra. This is applied to the category of Yetter-Drinfeld modules over a
Hopf group coalgebra, leading to the construction of the Drinfeld double. It is
shown that this Drinfeld double is a quasitriangular graded Hopf algebra.
