Cohomology characterizations of non-abelian extensions of Hom-Lie algebras

In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.