Non-abelian extensions and automorphisms of post-Lie algebras

In this paper, we introduce the concepts of crossed modules of post-Lie algebras and cat${}^1$-post-Lie algebras. It is proved that these two concepts are equivalent to each other. Secondly, we construct a non-abelian cohomology for post-Lie algebras to classify their non-abelian extensions. At last, we investigate the inducibility problem of a pair of automorphisms for post-Lie algebras and construct a Wells exact sequence to solve it.