  Let $1\to (K,K_1)\to (G,N_G(K_1))\to(Q,Q_1)\to 1$ be a short exact sequence
of pairs of finitely generated groups with $K$ strongly hyperbolic relative to
proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$
such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric
section $s\colon Q \to G$. Further we prove that if $G$ is strongly hyperbolic
relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative
to $K_1$, then there exists a Cannon-Thurston map for the inclusion
$i\colon\Gamma_K\to \Gamma_G$.
