Forcing hereditarily separable compact-like group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2c) which admit: a hereditarily separable group topology, a group topology making G into an S-space, 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), 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 2c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired). We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.