  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.
