Let $A$ be a commutative, cocommutative Hopfalgebra, finitely generated and projective overits base ring $R$. Waterhouse asked whether theimage of the class-invariant map, taking each $A$-Galois algebra to the class in ${\rm Pic}(A)$of its $R$-linear dual, is the group of primitiveclasses in ${\rm Pic}(A)$. We discuss functorialaspects of this problem, and relate it toFr\"ohlich's Hom-description of ${\rm Pic}(A)$in the case that $R$ is a Dedekind domain withfield of fractions $K$, and $A$ is an $R$-Hopforder in a separable $K$-Hopf algebra. We thenapply this machinery to a certain class of Hopf orders $\mathfrak A$ in the Hopf algebra${\rm Map}(G,K)$. More precisely, we give a positive answer to Waterhouse's question for$\mathfrak A$ when the dual $\mathfrak B$ of $\mathfrak A$ is one of the Hopf orders in$KG$ constructed by Larson, and a compatibilitycondition holds between the filtrations of $G$ determined by the various completions of $\mathfrak B$.

1991 Mathematics Subject Classification:11R33, 16W30