begun in the papers cited above, our results have applications in the study of the Hopf-Galois module structure of rings of algebraic integers in quaternionic extensions of local or global fields. Since such extensions have been important in the history of Galois module structure (see [@MAR1971], for example), this has the potential to be a fruitful line of inquiry, which we intend to pursue in a future paper.

A theorem of Greither and Pareigis ([@GRE1987 Theorem 3.1], see also [@CHI2000 Theorem 6.8]) classifies all of the Hopf-Galois structures admitted by a finite separable extension of fields. We state it here in a weakened form applicable to finite Galois extensions. Consider the group of permutations on the underlying set of $G$, ${\text{Perm}}(G)$, and let $ \lambda : G \hookrightarrow {\text{Perm}}(G) $ be the left regular representation. A subgroup $N$ of ${\text{Perm}}(G)$ is said to be *regular* if $|N|=|G|$, the stabiliser ${\text{Stab}}_N(g)=\{\eta\in N\,|\,\eta\cdot g=g\}$ is trivial for all $g\in G$, and $N$ acts transitively on $G$ (any two of these properties imply the third). The theorem of Greither and Pareigis states that there is a bijection between regular subgroups $N$ of ${\text{Perm}}(G)$ normalised by $\lambda(G)$ and Hopf Galois structures on $L/F$. Furthermore, 