Fully transitive, transitive Abelian groups and some their generalizations

In the study of Abelian groups, the fact that homomorphisms mapping subgroups of a group into this group can be extended to an endomorphism of the whole group is an important property of homomorphisms. For example, (fully) transitive torsion-free groups can be defined as groups in which all (homomorphisms) height-preserving homomorphisms from any pure rank $1$ subgroup into this group are extended to (endomorphisms) automorphisms of the group. In this paper, some equivalent feasibility conditions for a group to be (fully) transitive, endotransitive, or weakly transitive are given. Relations between these notions are also shown.
It is easy to show that a direct summand of a fully transitive group is a fully transitive group. There exist transitive $p$-groups which have a nontransitive direct summand. At the same time, the question whether the class of torsion free transitive groups is closed with respect to taking direct summands remains open. In this paper, some necessary and sufficient conditions under which a direct summand of an arbitrary transitive group is a transitive group are proposed.
There is a well-known Corner’s criterion on (full) transitivity of a reduced $p$-group. Below, this result is generalized to arbitrary reduced Abelian groups.