of $G$ on $\lambda(G)$ gives $G$-orbits equal to the conjugacy classes. When $G$ is not abelian (so that the $\rho(G)\neq\lambda(G)$) the structure for which $N=\lambda(G)$ is the underlying group is called the *canonical non-classical* structure.

The theorem of Greither and Pareigis is the cornerstone of almost all of the work concerned with the enumeration, description, and application of Hopf-Galois structures on separable extensions of fields. In particular, via a theorem of Byott [@BYOTT1996 Proposition 1], it reveals a connection between the theory of Hopf-Galois structures and the theory of [*left skew braces*]{}, which is described in detail in the appendix to [@BYO2018]. This appendix contains an enumeration of the Hopf-Galois structures admitted by a quaternion extension $ L/F $ [@BYO2018 Table A.1]. In section 2 below we compute the regular subgroups corresponding to these Hopf-Galois structures, and in section 3 we determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In section 4 we study the $ F $-algebra structure of these Hopf algebras: under the assumption that $ F $ has characteristic zero, we find explicit bases for each Hopf algebra, compute their Artin-Wedderburn decompositions, and identify which are isomorphic as $ F $-algebras.\
The 