We consider a class of quasi-Hopf algebras which we call \emph{generalized
twisted quantum doubles}. They are abelian extensions H = \mb{C}[\bar{G}]
\bowtie \mb{C}[G] (G is a finite group and Gˉ a homomorphic image),
possibly twisted by a 3-cocycle, and are a natural generalization of the
twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show
that if G is a subgroup of SU_2(\mb{C}) then H exhibits an orbifold McKay
Correspondence: certain fusion rules of H define a graph with connected
components indexed by conjugacy classes of Gˉ, each connected component
being an extended affine Diagram of type ADE whose McKay correspondent is the
subgroup of G stabilizing an element in the conjugacy class. This reduces to
the original McKay Correspondence when Gˉ=1.Comment: 5 figure