Let \(A\) be a group. If \(a_1,\ldots,a_n\in A\) then, when considering a model \((A,a_1,\ldots,a_n)\), we assume that the elements \(a_1,\ldots,a_n\) are distinguished as constants. If models \(A\) and \(B\) are elementarily equivalent then we write \(A\equiv B\). A countable group \(A\) is called homogeneous if, for arbitrary sequences of elements \(a_1,\ldots,a_{n+1}\) and \(b_1,\ldots,b_n\) in \(A\), the relation \((A,a_1,\ldots,a_n)\equiv(A,b_1,\ldots,b_n)\) implies existence of an element \(b_{n+1}\in A\) such that \((A,a_1,\ldots,a_{n+1})\equiv(A,b_1,\ldots,b_{n+1})\). The main results of the article are as follows: it is proven that a countable Abelian torsion-free group is homogeneous if and only if its reduced part is homogeneous; a homogeneity condition involving \(p\)-adic numbers is obtained for a group of some class of Abelian groups; and an example is given of a strongly constructive homogeneous group whose reduced part is homogeneous and not constructive.