Extensions and automorphisms of Lie algebras

Let $0 \to A \to L \to B \to 0$ be a short exact sequence of Lie algebras over a field $F$, where $A$ is abelian. We show that the obstruction for a pair of automorphisms in $\Aut(A) \times \Aut(B)$ to be induced by an automorphism in $\Aut(L)$ lies in the Lie algebra cohomology $\Ha^2(B;A)$. 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 $\Aut\big(L_{n,2}^{(1)}\big) \times \Aut\big(L_{n,2}^{ab}\big)$ to be induced by an automorphism in $\Aut\big(L_{n,2}\big)$, where $L_{n,2}$ is a free nilpotent Lie algebra of rank $n$ and step $2$.