On the Galois module structure of the square root of the inverse different in abelian extensions

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$ and $G$ a finite group of odd order. If $K_h$ is a weakly ramified $G$-Galois $K$-algebra, then its square root $A_h$ of the inverse different is a locally free ${\mathcal{O}}_{K}G$-module and hence determines a class in the locally free class group $Cl({\mathcal{O}}_{K}G)$ of ${\mathcal{O}}_{K}G$. We show that for $G$ abelian and under suitable assumptions, the set of all such classes is a subgroup of $Cl({\mathcal{O}}_{K}G)$.