C⁎-algebras of extensions of groupoids by group bundles

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