Abstract: Let $G$ be a hyperbolic group and $H$ be a hyperbolic subgroup
of $G$. If the embedding $H\to G$ extends continuously to a map between
the Gromov compactifications of the groups, this extension is called a
Cannon-Thurston map (CT). While it is known that not every hyperbolic
subgroup embedding in a hyperbolic group admits CT, over time the
existence of CT has been proven in many cases. We will start with a survey
of these results and move on to the following case where CT exists. Let
$1\to N \to G \stackrel{\pi}{\to} Q\to 1$ be a short exact sequence of
non-elementary hyperbolic groups and $K=\pi^{-1}(Q_1)$, where $Q_1$ is a
qi-embedded subgroup of $Q$. Then $K$ is hyperbolic and $K\to G$ admits
CT. This is part of joint work with Pranab Sardar.