Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions σ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of σ on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C∗-algebra of this twist is isomorphic to the full C∗-algebra of σ, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.