Relative Galois module structure of octahedral extensions

Let k be a number field, Ok its ring of integers and Cl(k) its class group. Let Γ be the symmetric (octahedral) group S4. Let be a maximal Ok-order in the semisimple algebra k[Γ] containing Ok[Γ], its locally free class group, and the kernel of the morphism induced by the augmentation . Let N/k be a Galois extension with Galois group isomorphic to Γ, and ON the ring of integers of N. When N/k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to ON the class of , denoted , in . We define the set of realizable classes to be the set of classes such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to Γ, and for which . In the present article, we prove that is the subgroup of provided that the class number of k is odd.