Equivariant maps which are self homotopy equivalences

The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact mat the most elementary homotopy theory of “simplicial sets on which a fixed Simplicial group H acts” is equivalent to the homotopy theory of “simplicial sets over the classifying complex WH”, and (ii) to use his (in §1) to prove a classification theorem for simplicial sets with an H-action, which provides classifying complexes for their equivariant maps which are self homotopy equivalences.