  Given $X$ a finite nilpotent simplicial set, consider the classifying
fibrations $$ X\to Baut_G^*(X)\to Baut_G(X),\qquad X\to Z\to Baut_{\pi}^*(X),
$$
  where $G$ and $\pi$ 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 $\pi_*(X)$.
  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
$Der^G L$ and $Der^{\pi} L$ of the Lie algebra $Der L$ of derivations of $L$
such that the geometrical realization of the sequences of cdgl morphisms
  $$
  L\stackrel{ad}{\to} Der^G L\to Der^G L\widetilde\times sL,\qquad L\to
L\widetilde\times Der^{\pi} L\to Der^{\pi} L
  $$
  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 $\pi$ together with the rational homotopy type of the classifying
spaces $BG $ and $B\pi$.
