C*-Algebras of extensions of groupoids by group bundles

Given a normal subgroup bundle $\mathcal A$ of the isotropy bundle of a groupoid $\Sigma$, we obtain a twisted action of the quotient groupoid $\Sigma/\mathcal A$ on the bundle of group $C^*$-algebras determined by $\mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(\Sigma)$. Restricting to the case where $\mathcal A$ is abelian, we describe $C^*(\Sigma)$ as the $C^*$-algebra associated to a $\mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $\Sigma/\mathcal A$ on the Pontryagin dual space of $\mathcal A$. We give some illustrative examples of this result.