  Given a metric (graph) bundle $X$ over $B$ where all the fibres are strongly
relatively hyperbolic and nonelementary we show that, under certain conditions,
$X$ is strongly hyperbolic relative to a collection of maximal cone-subbundles
of horosphere-like spaces. Further, given a coarsely Lipschitz qi embedding $i:
A\to B$, we show that the pullback $Y$ is strongly relatively hyperbolic and
the map $Y\to X$ admits a Cannon-Thurston (CT) map. As an application, we prove
a group-theoretic analogue of this result for a relatively hyperbolic extension
of groups.
