Complementing a subgroup of a hyperbolic group by a free factor

Let $G$ be a hyperbolic group that is not almost cyclic and $H$ be its quasiconvex subgroup of infinite index. We find necessary and sufficient conditions of there being for $H$ a free subgroup $F$ of rank 2 in $G$ such that $F$ and $H$ generate a free product $F*H\subseteq G$. It is proved that $F*H$ is quasiconvex and that there exists an algorithm for verifying the conditions of the criterium given $G$ and $H$.