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 OKOK and let G be a finite group of odd order. Given a G-Galois K  -algebra KhKh, let AhAh denote its square root of the inverse different, which exists by Hilbert's formula. If Kh/KKh/K is weakly ramified, then a result of Erez implies that AhAh is locally free over OKGOKG and hence defines a class in the locally free class group Cl(OKG)Cl(OKG) of OKGOKG. In the case that G   is abelian, we study the collection of all such classes and show that a subset of them is in fact a subgroup of Cl(OKG)Cl(OKG).