Hopf-Galois structures on separable field extensions of degree $pq$

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions $L\mathrm{/}K$ is a natural next step. One must consider now the interplay between two Galois groups $G=\mathrm{Gal}(E\mathrm{/}K)$ and $G^{\mathrm{\prime }}=\mathrm{Gal}(E\mathrm{/}L)$, where $E$ is the Galois closure of $L\mathrm{/}K$. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree $pq$ where $p$ and $q$ are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that $p=2q+1$.