Localization in equivariant cohomology theory for groups other than ($p$-)tori

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Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups:

Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite cohomological dimension, and $F$ the fixed point set of $X$. Then the following localized restriction homomorphism $$ S^{-1} H^*_G(X;\mathbb{Q}) \to S^{-1}H^*_G(F;\mathbb{Q}), $$ where $S = \mathbb{Q}[t_1, \ldots, t_k] - \{0\}$, is an isomorphism.

Something similar happens when $G$ is an elementary abelian $p$-group.

On the other hand, Hsiang points out in a couple of places in his book (e.g. top of p. 44, Remark (i) on p. 68) that those mentioned above the only classes of groups for which one can expect such a theorem to work. He presents an example that it is indeed unreasonable if we consider compact connected non-abelian Lie groups.

My question is: what about other groups, e.g. products $G = (S^1)^k \times (\mathbb{Z}_p)^l$? Does this sort of groups admit a localization theorem as above, perhaps for cohomology with $\mathbb{Z}$ coefficients?