We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property $\h$). Every pseudocompact Abelian group $G$ with cardinality $|G|\leq 2^{2^\cc}$ satisfies this inequality and therefore admits a pseudocompact group topology with property $\h$. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property $\h$. We also observe that pseudocompact Abelian groups with property $\h$ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.

19 pages; In this version we work assuming SCH (Singular Cardinal Hypothesis), whereas in our previous version we had to assume GCH (Generalized Continuum Hypothesis). The general problem is still open in ZFC, but models avoiding SCH are much harder to come by. We thank professors W. W. Comfort and D. Dikranjan for their help concerning Example 5.9.