  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 of odd order, 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 a partial analogue of a classical theorem of Shafarevich on the
inverse Galois problem for soluble groups in the setting of Galois module
theory.
