Forcing hereditarily separable countably compact group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2^c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2^c) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), (iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary). As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2^c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired).