The author proves the following theorem for a wide class of groups including finite groups, compact Lie groups, groups of finite virtual cohomological dimension over \(\mathbb{F}_p\) and profinite groups. Theorem. Let \(f\colon H\to G\) be a homomorphism such that the induced map \(f^*\colon H^*(G;\mathbb{F}_p)\to H^*(H;\mathbb{F}_p)\) is an isomorphism in high degrees. Then \(f\) induces an equivalence of categories \(\mathcal S_p(H)\to\mathcal S_p(G)\), where \(\mathcal S_p\) denotes the category of \(p\)-subgroups with the morphisms induced by inclusion and conjugation. This was proved by \textit{G. Mislin} for compact Lie groups [Comment. Math. Helv. 65, No. 3, 454-461 (1990; Zbl 0713.55009)] and by \textit{C.-N. Lee} for groups of finite virtual cohomological dimension [Topology 33, No. 4, 721-728 (1994; Zbl 0823.55013)]. These proofs used some deep topological methods. Here the author takes a more algebraic approach, which enables him to prove the theorem for groups of finite virtual cohomological dimension and profinite groups. The proof uses \textit{J. Lannes}'s \(T\)-functor [Cohomology of groups and function spaces, unpublished notes (1986)].