Suppose that $K \subset G$ is a topological subgroup and this inclusion is a homotopy equivalence (so somewhat stronger than what's written in the title). I'm not assuming compactness but am happy to assume they're Lie groups.
Is there a relationship between their equivariant cohomologies? That is, we have $H^*_G(X):=H^*_{\text{sing}}(X \times_G EG)$ and a similar definition for $H^*_K(X)$. The best situation is that they're isomorphic but that's probably false in general.