  Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$
be a \mbox{finite group of} odd order. Given a $G$-Galois $K$-algebra $K_h$,
let $A_h$ be the square root of the inverse different of $K_h/K$, which exists
by Hilbert's formula. If $K_h/K$ is weakly ramified, then $A_h$ is locally free
over $\mathcal{O}_{K}G$ by a result of B. Erez, in which case it determines a
class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of
$\mathcal{O}_KG$. Such a class in $\mbox{Cl}(\mathcal{O}_KG)$ is said to be
$A$-realizable, and tame $A$-realizable if $K_h/K$ is tame. Let
$\mathcal{A}(\mathcal{O}_KG)$ and $\mathcal{A}^t(\mathcal{O}_KG)$ denote the
sets of all $A$-realizable classes and tame $A$-realizable classes,
respectively. For $G$ abelian, we will show that the two sets
$\mathcal{A}(\mathcal{O}_KG)$ and $\mathcal{A}^t(\mathcal{O}_KG)$ are equal
when extended scalars to the maximal order in $KG$.
