Absolute closedness of torsion-free Abelian groups in the class of metabelian groups

The dominion of a subgroup $H$ of a group $G$ in a class $M$ is the set of all elements $a\in G$ whose images are equal for all pairs of homomorphisms from $G$ to each group in $M$ that coincide on $H$. A group $H$ is absolutely closed in a class $M$ if, for any group $G$ in $M$, every inclusion $H\le G$ implies that the dominion of $H$ in $G$ (in $M$) coincides with $H$.
We deal with dominions in torsion-free Abelian subgroups of metabelian groups. It is proved that every nontrivial torsion-free Abelian subgroup is not absolutely closed in the class of metabelian groups. It is stated that if a torsion-free subgroup $H$ of a metabelian group $G$ and the commutator subgroup $G'$ have trivial intersection, then the dominion of $H$ in $G$ (in the class of metabelian groups) coincides with $H$.