Quasi-invariant lifts of completely positive maps for groupoid actions

Let G be a locally compact, -compact, Hausdorff groupoid and A be a separable, C₀ (G^ (0) ) -nuclear, G-C^*-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from A into a separable, quotient C^*-algebra. Along the way, we construct the Busby invariant for G-actions.