Realizable classes and embedding problems

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$ and let $G$ be a finite group. Given a $G$-Galois $K$-algebra $K_h$, let ${\mathcal{O}}_h$ denote its ring of integers. If $K_h\mathrm{/}K$ is tame, then a classical theorem of E. Noether implies that ${\mathcal{O}}_h$ is locally free over ${\mathcal{O}}_{K}G$ and hence defines a class in the locally free class group of ${\mathcal{O}}_{K}G$. For $G$ abelian, by combining the work of J. Brinkhuis and L. McCulloh, we prove that the structure of the collection of all such classes is related to the study of embedding problems.