In this paper, we study the theory of non-abelian extensions of a Leibniz conformal algebra $R$ by a Leibniz conformal algebra $H$ and prove that all the non-abelian extensions are classified by non-abelian $2$nd cohomology $H^2_{ nab}(R,H)$ in the sense of equivalence. Then we introduce a differential graded Lie algebra $\mathfrak{ L}$ and show that the set of its Maurer-Cartan elements in bijection with the set of non-abelian extensions. Finally, as an application of non-abelian extension, we consider the inducibility of a pair of automorphisms about a non-abelian extension, and give the fundamental sequence of Wells of Leibniz conformal algebra $R$. Especially, we discuss the extensibility problem of derivations about an abelian extension of $R$.
Categories: math.RA