In two separate pieces of 1970's work, Fred Cohen first computed the mod p homology of C_nX (and \Omega^n \Sigma^n X) as functors of the homology of X and later proved (with Larry Taylor and myself) that the suspension spectrum of C_nX splits as the wedge of the suspension spectra of its filtration quotients D_{n,j} X. For a representation V of a finite group G, one can ask for equivariant analogues of these statements, considering C_V X and its filtration quotiients D_{V,j}X. I conjecture that the equivariant homology statement is true, although the question is still open even for the cyclic group C_2. I will explain that the equivariant splitting statement is true for any finite group G and representation V, by updated mimicry of the proof of Fred, Larry, and myself.