If two spaces $X$ and $Y$ are homotopy equivalent, then their cohomologies are isomorphic $$H(X) \cong H(Y).$$ Is there a similar result for the equivariant cohomology? Given algebraic varieties $X_1, X_2$ with actions of $G_1,G_2$, respectively. If $X_1$ and $X_2$ are homotopy equivalent, as well as $G_1$ and $G_2$, can we say that $$H^*_{G_1}(X_1) \cong H^*_{G_2}(X_2)?$$
I am trying to understand this paper: https://arxiv.org/pdf/1006.2706.pdf
On page 9, there is this sequence of three morphisms. The second one is clear, but the others are not. The first one is induced by the homotopy equivalence of the spaces, but I don't see how that works. Can anyone help? Further, the third morphism is unclear too. How does the pushforward work when we change the group?
Thanks in advance!