A space X is selectively sequentially pseudocompact if for every family {U-n : n is an element of N} of non-empty open subsets of X, one can choose a point x(n) is an element of U-n for every n is an element of N in such a way that the sequence {x(n) : n is an element of N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of Garcia-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of Garcia-Ferreira and Tomita. (C) 2017 Elsevier B.V. All rights reserved.