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 M be a maximal Ok-order in the semisimple algebra k[Γ] containing Ok[Γ], Cl(M) its locally free class group, and Cl○(M) the kernel of the morphism Cl(M)→Cl(k) induced by the augmentation M→Ok. 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 M⊗Ok[Γ]ON, denoted [M⊗Ok[Γ]ON], in Cl(M). We define the set R(M) of realizable classes to be the set of classes c∈Cl(M) such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to Γ, and for which [M⊗Ok[Γ]ON]=c. In the present article, we prove that R(M) is the subgroup Cl○(M) of Cl(M) provided that the class number of k is odd.