Quillen proved in 1971 ("The Spectrum of an Equivariant Cohomology Ring: I,II") for a large class of groups $G$ including

• compact Lie groups
• groups of finite virtual cohomological dimension
• compact topological groups with a finite number of conjugacy classes of elementary abelian $p$-subgroups

that the map $$\text{res}: H^\ast(BG,\mathbb{F}_p) \to \varprojlim_E H^\ast(BE,\mathbb{F}_p)$$ where $E$ runs through the elementary abelian $p$-subgroups of $G$ (ordered by inclusion and conjugacy) is an F-isomorphism (i.e. it has finite kernel and for each $x$ in the RHS, a $p$-power of $x$ is in the image).

Question 1: Has this theorem been generalized to other classes of groups in the meantime ?

Conversely, I'm also interested in counterexamples to the theorem, i.e.

Question 2: What is an example of a topological group $G$ s.t. the map above is no F-isomorohism ?

Quillen proved in 1971 ("The Spectrum of an Equivariant Cohomology Ring: I,II") for a large class of groups $G$ including

• compact Lie groups
• groups of finite virtual cohomological dimension
• compact topological groups with a finite number of conjugacy classes of elementary abelian $p$-subgroups

that the map $$\text{res}: H^\ast(BG,\mathbb{F}_p) \to \varprojlim_E H^\ast(BE,\mathbb{F}_p)$$ where $E$ runs through the elementary abelian $p$-subgroups of $G$ (ordered by inclusion and conjugacy) is an F-isomorphism (i.e. it has finite kernel and for each $x$ in the RHS, a $p$-power of $x$ is in the image).

Question 1: Has this theorem been generalized to other classes of groups in the meantime ?

Conversely, I'm also interested in counterexamples to the theorem, i.e.

Question 2: What is an example of a topological group $G$ s.t. the map above is no F-isomorphism ?