On Kneser Extensions

LetE/Fbe a finite separable extension such thatE = F(G) withF* ? G ? E*,G/F* a finite group, and such that there exists a 11 correspondence between the subfields of the extensionE/Fand the subgroups ofG/F*, butE/Fis notG-cogalois. A necessary and sufficient condition is given for the existence of another groupG?for the extensionE/Fto beG?-cogalois.