Rational division algebras as solvable crossed products

LetG be a finite group. If there exists a division algebra central over the rationalsQ which is a crossed product forG, then according to a theorem of Schacher, the Sylow subgroups ofG are all metacyclic. The converse is proved here to hold in the following cases: (1)G metacyclic; (2) The Sylow 2-subgroups ofG are cyclic (this impliesG solvable); (3)G is solvable and the Sylow 2-subgroups ofG are dihedral of order larger than 8.