Fully invariant subgroups of Abelian groups and full transitivity

An Abelian group $A$ is said to be fully transitive if for any elements $a,b\in A$ with $\mathbb H(a)\leqslant\mathbb H(b)$ ($\mathbb H(a)$, $\mathbb H(b)$ are the height-matrices of elements $a$ and $b$) there exists an endomorphism of $A$ sending $a$ into $b$. We say that an Abelian group $A$ is $\mathbb H$-group if any fully invariant subgroup $S$ of $A$ has the form $S=\{a\in A\mid\mathbb H(a)\geqslant M\}$, where $M$ is some $\omega\times\omega$-matrix with ordinal numbers and symbol $\infty$ for entries. The description of fully transitive groups and $\mathbb H$-groups in various classes of Abelian groups is obtained. The results of this paper show that every $\mathbb H$-group is a fully transitive group, but there are fully transitive torsion free groups and mixed groups, which are not $\mathbb H$-groups. The full description of fully invariant subgroups and their lattice for fully transitive groups in various classes of Abelian groups is obtained.