On the relative Galois module structure of rings of integers in tame extensions

Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work of the second-named author in the context of relative algebraic $K$ theory. When $G$ is nilpotent, we show (subject to certain conditions) that the set of realisable classes is a subgroup of $Cl(O_FG)$. This may be viewed as being an analogue of a classical theorem of Scholz and Reichardt on the inverse Galois problem for nilpotent groups in the setting of Galois module theory.