Groups of order 4p, twisted wreath products and Hopf–Galois theory

The work of Greither and Pareigis details the enumeration of the Hopf–Galois structures (if any) on a given separable field extension. We consider the cases where L/K is already classically Galois with Γ=Gal(L/K), where Γ=4p for p>3 a prime. The goal is to determine those regular (transitive and fixed point free) subgroups N of Perm(Γ) that are normalized by the left regular representation of Γ. A key fact that aids in this search is the observation that any such regular subgroup, necessarily of order 4p, has a unique subgroup of order p. This allows us to show that all such N are contained in a ‘twisted’ wreath product, a subgroup of high index in Perm(Γ) which has a very computationally convenient description that allows us to perform the aforementioned enumeration.