Automorphisms of fibrations

Let X be a simplicial set, G a simplicial group and WG the classifying complex of G. Then it is well known [1], [3] that the principal fibrations with base X and group G are classified by the components of the function complex (WG)^X. The aim of the present note is to prove the following complement to this result (1.2): Let p be a principal fibration with base X and group G, and let aut p be its simplicial group of automorphisms (which keep the base fixed). Then W(aut p) has the homotopy type of the component of (WG)^X which (see above) corresponds to p. A similar result holds for ordinary fibrations.