Hopf–Galois structures on parallel extensions

Let L/K be a finite separable extension of fields of degree n, and let E/K be its Galois closure. Greither and Pareigis showed how to find all Hopf–Galois structures on L/K. We will call a subextension L^′/K of E/K parallel to L/K if [L^′:K]=n. In this paper, we investigate the relationship between the Hopf–Galois structures on an extension L/K and those on the related parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf–Galois structure but that has a parallel extension admitting no Hopf–Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree pq with p,q distinct odd primes, and show that there is no example of such an extension admitting the phenomenon.