Let a compact group $G$ act on a manifold $M$. Why does G act as a group of automorphisms on $H^p (M, \mathbb{R})$?

Question

Let $(\sigma , m) \rightarrow t_{\sigma}$ be this action. A definition for acting on a group of automorphisms I found here: Action via automorphism. t^{*}_{\sigma} seems to be this action and I can prove, it is an action, but do not how to prove the property making it an action of automorphims, because I do not know how the group action of the de Rham cohomology works / how to work with it.