In this paper, we introduce the notions of crossed module of anti-pre-Lie algebras. We explore the non-abelian cohomology of anti-pre-Lie algebras to classify their non-abelian extensions. Additionally, we investigate the inducibility of a pair of automorphisms for anti-pre-Lie algebras and construct a Wells exact sequence. Finally, we show how these results fit with abelian extensions of anti-pre-Lie algebras.