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 de Rham cohomology of a Lie groupoid by considering de Rham cohomology of simplicial manifold associated to it, denoted by $\mathcal{G}_\bullet$.

Is there any relation between de Rham cohomology of Lie groupoid $\mathcal{G}$ (eg page 11 of Laurent-Gengoux–Tu–Xu, *Chern-Weil map for principal bundles over groupoids*, Math. Z. **255** (2007) pp451–491, arXiv:math/0401420) and the de Rham (??) cohomology of the Lie group $G$ and the de Rham cohomology of the manifold $M$?