Let be a short exact sequence of Lie algebras over a field , where is abelian. We show that the obstruction for a pair of automorphisms in to be induced by an automorphism in lies in the Lie algebra cohomology . As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in Aut Ln,2(1) ×Aut L n,2ab to be induced by an automorphism in Aut Ln,2, where is a free nilpotent Lie algebra of rank and step .