  Let G be a finite group of exponent m and let k be a field of characteristic
prime to m, containing the m-th roots of unity. For any Rost cycle module M
over k, we construct exact sequences detecting the unramified elements in
Serre's group of invariants of G with values in M in terms of "residue"
morphisms associated to pairs (D,g), where D runs through the subgroups of G
and g runs through the homomorphisms \mu_m \to G whose image centralises D.
This allows us to recover results of Bogomolov and Peyre on the unramified
cohomology of fields of invariants of G, and to generalise them.
