||Suppose N is a nice subgroup of the primary abelian group G and A = G/N. The paper discusses various contexts in which G satisfying some property implies that A also satisfies the property, or visa versa, especially when N is countable. For example, if n is a positive integer, G has length not exceeding ω1 and N is countable, then G is n-summable iff A is n-summable. When A is separable and N is countable, we discuss the condition that any such G decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that A must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups.