Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

We study the problem of finite approximability with respect to conjugacy of amalgamated free products by a normal subgroup and prove the following assertions. A) If $G$ is the amalgamated free product $G=G_1*_HG_2$ of polycyclic groups $G_1$ and $G_2$ by a normal subgroup $H$, where $H$ is an almost free Abelian group of rank 2, then $G$ is finitely approximate with respect to conjugacy. B) (i) If $G_1=G_2=L$ is a polycyclic group and $G=G_1*_HG_2$ is the amalgamated product of two copies of the group $L$ by a normal subgroup $H$, then $G$ is finitely approximable with respect to conjugacy. (ii) If $G$ is an amalgamated free product $G=G_1*_HG_2$ of polycyclic groups $G_1$ and $G_2$ by a normal subgroup $H$, where $H$ is central in $G_1$ or $G_2$, then $G$ is finitely approximable with respect to conjugacy.