  In this paper, we explore non-abelian extensions of relative Rota-Baxter Lie
algebras and classify the non-abelian extensions by introducing the non-abelian
second cohomology group. We also study the inducibility of a pair of
automorphisms about a non-abelian extension of relative Rota-Baxter Lie
algebras and derive the Wells type exact sequences. Finally, we investigate the
inducibility problem of pairs of derivations about an abelian extension of
relative Rota-Baxter Lie algebras and give an exact sequence of Wells type.
