The Dixmier-Douady Classes of Certain Groupoid $C^{\ast }$-Algebras with Continuous Trace

Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier–Douady invariant of the $C^{\ast }$-algebra of the groupoid extension associated to a \v{C}ech $2$-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 'etale equivalence relations.