On a Family of Residually Finite Groups

As is known, there is a finitely generated residually finite group (for short, a residually $\mathcal F$-group) whose extension by some finite group is not a residually $\mathcal F$-group. In the paper, it is shown that, nevertheless, every extension of a finite group by a finitely generated residually $\mathcal F$-group is a Hopf group, and every extension of a center-free finite group by a finitely generated residually $\mathcal F$-group is a residually $\mathcal F$-group. If a finitely generated residually $\mathcal F$-group $G$ is such that every extension of an arbitrary finite group by $G$ is a residually $\mathcal F$-group, then a descending HNN-extension of the group $G$ also has the same property, provided that it is a residually $\mathcal F$-group.