Obstructions to lifting cocycles on groupoids and the associated $C^{\ast }$-algebras

Given a short exact sequence of locally compact abelian groups $0 \to A \to B \to C \to 0$andacontinuous$C$-valued$1$-cocycle$\phi$ on a locally compact Hausdorff groupoid $\mathrm{\Gamma }$ we construct a twist of $\mathrm{\Gamma }$ by $A$ that is trivial if and only if $\varphi$ lifts. The cocycle determines a strongly continuous action of $\widehat{C}$ into $\mathrm{Aut}{C}^{\ast }(\mathrm{\Gamma })$ and we prove that the $C^{\ast }$-algebra of the twist is isomorphic to the induced algebra of this action if $\mathrm{\Gamma }$ is amenable. We apply our results to a groupoid determined by a locally finite cover of a space $X$ and a cocycle provided by a \v{C}ech 1-cocycle with coefficients in the sheaf of germs of continuous $\mathbb{T}$-valued functions. We prove that the $C^{\ast }$-algebra of the resulting twist is continuous trace and we compute its Dixmier-Douady invariant.