Let $\Gamma$ be an amenable locally compact groupoid and let $A$ be a closed subgroup of a locally compact abelian group $B$. Given a $B/A$-valued 1-cocycle $\phi$ on $\Gamma$, there is a central extension $\Sigma_\phi$ of $\Gamma$ by $A$ which is trivial iff $\phi$ lifts to a $B$-valued cocycle. We prove that $C^*(\Sigma_\phi)$ is isomorphic to the induced algebra of the natural action of $(B/A)^\hat$ on $C^*(\Gamma)$. We also consider a simple class of examples arising from Cech 1-cocycles.