Higher cyclic reduced powers

Let denote the cyclic construction on the space X of order n=2k. For any k≥2 we construct an endofunctorCn on the category of unstable modules over the (mod 2) Steenrod algebra, such that there is a natural isomorphism Cn(H*X) H*(CnX). We also find explicit formulas for the cohomology of the diagonal map BCn×X→CnX. Its linearity over the Steenrod algebra is explained by relations similar to the Adem relations. All may be seen as a generalization of the classical case k=1.