 de Rham cohomology of a finite quotient

For a free smooth action of a finite group $G$ on a smooth manifold $M$, the quotient projection inducesInvariant differential forms descend uniquely through local inverse branches of the covering map. Averaging by $|G{|}^{-1}{\sum }_{g\in G}{g}^{\ast }$ commutes with the exterior derivative, produces invariant representatives of invariant classes, and produces an invariant primitive for every invariant exact form. Hence taking invariant forms and taking invariant cohomology give the same result over the real numbers.