  In this short note we answer to a question of group theory from
arXiv:0910.5080. In that paper the author describes the set of realizable
Steinitz classes for so-called $A'$-groups of odd order, obtained iterating
some direct and semidirect products. It is clear from the definition that
$A'$-groups are solvable $A$-groups, but the author left as an open question
whether the converse is true. In this note we prove the converse when only two
prime numbers divide the order of the group, but we show it to be false in
general, producing a family of counterexamples which are metabelian and with
exactly three primes dividing the order. Steinitz classes which are realizable
for such groups in the family are computed and verified to form a group.
