  $H$ is called a $G$-subgroup of a hyperbolic group $G$ if for any finite
subset $M\subset G$ there exists a homomorphism from $G$ onto a non-elementary
hyperbolic group $G_1$ that is surjective on $H$ and injective on $M$. In his
paper in 1993 A. Ol'shanskii gave a description of all $G$-subgroups in any
given non-elementary hyperbolic group $G$. Here we show that for the same class
of $G$-subgroups the finiteness assumption on $M$ (under certain natural
conditions) can be replaced by an assumption of quasiconvexity.
