Quasi-invariant lifts of completely positive maps for groupoid actions

Let $G$ be a locally compact, Hausdorff, second countable groupoid and $A$ be a separable, $C_0(G^{(0)})$-nuclear, $G$-$C^{\ast }$-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 $C^{\ast }$-algebra. Along the way, we construct the Busby invariant for $G$-actions.