  Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle
g \rangle$, we define a drilling of $G$ along $g$, which is a relatively
hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied
procedure of drilling a hyperbolic 3-manifold along an embedded geodesic. We
prove that, under suitable conditions, a hyperbolic group with $2$-sphere
boundary admits a drilling where the resulting relatively hyperbolic group pair
$(\widehat{G}, P)$ has relatively hyperbolic boundary $S^2$. This allows us to
reduce the Cannon Conjecture (in the residually finite case) to a relative
version.
