  We force a classification of all the Abelian groups of cardinality at most
$2^\mathfrak c$ that admit a countably compact group with a non-trivial
convergent sequence. In particular, we answer (consistently) Question 24 of
Dikranjan and Shakhmatov for cardinality at most $2^{\mathfrak c}$, by showing
that if a non-torsion Abelian group of size at most $2^\mathfrak c$ admits a
countably compact Hausdorff group topology, then it admits a countably compact
Hausdorff group topology with non-trivial convergent sequences.
