Fell bundles associated to groupoid morphisms

Given a continuous open surjective morphism $\pi :G\rightarrow H$ of étale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its $C^*$-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of Fell concerning $C^*$-algebraic bundles over groups. The case $H=X$, a locally compact space, was treated earlier by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to a crossed product, the $C^*$-algebra of a Fell bundle arising from an action of the groupoid $H$ on a $C^*$-bundle over $H^0$. We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles