  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.
