The Dixmier-Douady classes of certain groupoid C*-algebras with continuous trace

Given a locally compact abelian group G, we give an explicit formula for the Dixmier-Douady invariant of the C*-algebra of the groupoid extension associated to a Ĉech 2-cocycle in the sheaf of germs of continuous G-valued functions. We then exploit the blow-up construction for groupoids to extend this to some more general central extensions of étale equivalence relations.