Given X a finite nilpotent simplicial set, consider the classifying fibrationsX→BautG⁎(X)→BautG(X)andX→Z→Bautπ⁎(X) where G and π denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of X which act nilpotently on H⁎(X) and π⁎(X). We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if L is a cdgl model of X, there are connected sub cdgl's DerGL and DerΠL of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphismsL→adDerGL→DerGLטsLandL→LטDerΠL→DerΠL have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev Q-completion of G and π together with the rational homotopy type of the classifying spaces BG and Bπ.