Frattini closed groups and adequate extensions of global fields

Let L be a finite Galois extension of a global field F. It is shown that if the Galois group G = Gal(L/F) satisfies a certain condition, then L is a maximal commutative subfield of some F-division algebra if and only if the intermediate field corresponding to the Frattini subgroup of G is also a maximal commutative subfield of some F-division algebra. In particular this condition holds if G is a supersolvable group.