On residual separability of subgroups in split extensions

In 1973, Allenby and Gregoras proved the following statement. Let $G$ be a split extension of a finitely generated group $A$ by the group $B$. 1) If in groups $A$ and $B$ all subgroups (all cyclic subgroups) are finitely separable, then in group $G$ all subgroups (all cyclic subgroups) are finitely separable; 2) if in group $A$ all subgroups are finitely separable, and in group $B$ all finitely generated subgroups are finitely separable, then in group $G$ all finitely generated subgroups are finitely separable. Recall that a group $G$ is said to be a split extension of a group $A$ by a group $B$, if the group $A$ is a normal subgroup of $G$, $B$ is a subgroup of $G$, $G=AB$ and $A\cap B = 1$. Recall also that the subgroup $H$ of a group $G$ is called finitely separable if for every element $g$ of $G$, which does not belong to the subgroup $H$, there exists a homomorphism of $G$ on a finite group in which the image of an element $g$ does not belong to the image of the subgroup $H$. In this paper we obtained a generalization of the Allenby and Gregoras theorem by replacing the condition of the finitely generated group $A$ by a more general one: for any natural number $n$ the number of all subgroups of the group $A$ of index $n$ is finite. In fact, under this condition we managed to obtain a necessary and sufficient condition for finite separability of all subgroups (of all cyclic subgroups, of all finitely generated subgroups) in the group $G$.