Relative Hyperbolic Extensions of Groups and Cannon-thurston Maps

Let 1 → (K, K1) → (G, NG(K1)) → (Q, Q1) → 1 be a short exact sequence of pairs of finitely generated groups with K strongly hyperbolic relative to the subgroup K1. Let relative hyperbolic boundary of K with respect to K1 contains atleast three distinct parabolic end points and for all g ∈ G there exists k ∈ K such that gK1g −1 = kK1k , then we prove that there exists a quasi-isometric section s : Q → G. Further we prove that if G is strongly hyperbolic relative to the normalizer subgroup NG(K1) and weakly hyperbolic relative to K1, then there exists a CannonThurston map for the inclusion i : ΓK → ΓG.