Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers Ok, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers ON of N determines a class in the locally free class group Cl(Ok[Γ]). We show that the set of classes in Cl(Ok[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(Ok[Γ]) to the ideal class group Cl(Ok), provided that the ray class group of Ok for the modulus 4Ok has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[Γ].