Non-abelian Extensions of Lie algebras with derivations

In this paper, we investigate non-abelian extensions of Lie algebras with derivations using several different approaches. We show that the theory of non-abelian extensions of a Lie algebra with a derivation can be characterized by means of the second non-abelian cohomology, the Deligne groupoid, the homotopy category of strict Lie $2$-algebras with strict derivations, and the notion of a $(\g, D)$-kernel, respectively. Moreover, within this unified framework, we address the following existence problem: given a non-abelian extension of Lie algebras $$\begin{CD} 0@>>>\h@>i>>\hat{\g}@>p>>\g @>>>0, \end{CD}$$ let $(K,D)\in\Der(\h)\times\Der(\g)$ be a pair of derivations of $\h$ and $\g$ respectively. When does there exist a derivation $\hat{D}$ of $\hat{\g}$ such that $\hat{D}|_\h=K$ and $D\circ p=p\circ\hat{D}.$ We provide an obstruction class for the existence of such a lift.