Summary: Let \(k\) be a number field with ring of integers \(\mathfrak{O}_k\), 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 \(\mathfrak{O}_{N}\) of \(N\) determines a class in the locally free class group \(\mathrm{Cl}(\mathfrak{O}_k[\Gamma])\). We show that the set of classes in \(\mathrm{Cl}(\mathfrak{O}_k[\Gamma])\) realized in this way is the kernel of the augmentation homomorphism from \(\mathrm{Cl}(\mathfrak{O}_k[\Gamma])\) to the ideal class group \(\mathrm{Cl}(\mathfrak{O}_k)\). This refines a result of M. Godin and B. Sodaïgui [J. Number Theory 98, No. 2, 320–328 (2003; Zbl 1028.11067)] on the 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 \(\mathrm{Cl}(\mathfrak{O}_k[\Gamma])\) has been determined for a nonabelian group \(\Gamma\) and an arbitrary number field \(k\).