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Abstract.
Suppose $A$ is an abelian torsion group
with a subgroup $G$ such that
$A/G$ is countable that is, in other words, $A$ is a torsion
countable abelian
extension of $G$. A problem of some group-theoretic interest
is that of whether $G \in \mathbb K$, a class of abelian groups,
does imply that $A\in \mathbb K$. The aim of the present paper is to
settle the
question for certain kinds of groups, thus extending a classical result due
to Wallace (J. Algebra, 1981) proved when $\mathbb K$ coincides with the
class of all totally projective $p$-groups.
AMSclassification. 20K10, 20K20, 20K21.
Keywords. Countable factor-groups, $\Sigma$-groups, $\sigma$-summable groups, summable groups, $p^{\omega + n}$-projective groups.