to a family of p-permutation modules for kNG(Q)/Q, where Q runs over the non-trivial p-subgroups of G, together with certain isomorphisms. Conversely, the data of a compatible family of kNG(Q)/Q-modules comes from a

Abstract. Let F denote the class of finite groups, and let P denote the subclass consisting of groups of prime power order. We study group actions on topological spaces in which either (1) all stabilizers lie in P or (2) all stabilizers lie in F. We compare the classifying spaces for actions with stabilizers in F and P, the Kropholler hierarchies built on F and P, and group cohomology relative to F and to P. In terms of standard notations, we show that F ⊂ H1P ⊂ H1F, with all inclusions proper; that HF = HP; that FH ∗ (G; −) = PH ∗ (G; −); and that EPG is finite-dimensional if and only if EFG is finite-dimensional and every finite subgroup of G is in P. 1.