Let $G$ be a Lie group acting on a manifold $M$. 

Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$.  We have the notion of deRham cohomology of a Lie groupoid by considering deRham cohomology of simplicial manifold associated to it, denoted by $\mathcal{G}_\bullet$.

Is there any relation between deRham cohomology of Lie groupoid $\mathcal{G}$ and the deRham(??) cohomology of the Lie group $G$ and the deRham cohomology of the manifold $M$?