Realizable Galois module classes for tetrahedral extensions

Let k be a number field with ring of integers D-k, and let Gamma = A(4) be the tetrahedral group. For each tame Galois extension N/k with group isomorphic to Gamma, the ring of integers D-N of N determines a class in the locally free class group Cl(D-k[Gamma]). We show that the set of classes in Cl(D-k[Gamma]) realized in this way is the kernel of the augmentation homomorphism from Cl(D-k[Gamma]) to the ideal class group Cl(D-k). This refines a result of Godin and Sodaigui (J. Number Theory 98 (2003), 320-328) on Galois module structure over a maximal order in k[Gamma]. To the best of our knowledge, our result gives the first case where the set of realizable classes in Cl(D-k[Gamma]) has been determined for a nonabelian group Gamma and an arbitrary number field k.