if $ s \neq s^{\prime} $ then there cannot be a $ G $-equivariant isomorphism between $ D_{s,\lambda} $ and $ D_{s^{\prime}, \lambda} $.

Finally, there cannot be a $ G $-equivariant isomorphism between $ D_{s,\lambda} $ and $ D_{s^{\prime}, \rho} $ for any $ s,s^{\prime} $, since these groups have different numbers of $ G $-orbits.

These results agree with the number of isomorphism classes of Hopf algebras for each type given in Table 1 of [@CRE2017]. It may also be worth noting that our results imply that the Hopf-Galois structures of abelian type occur in pairs, with each pair arising from two different actions of a single Hopf algebra, and that, by contrast, each Hopf-Galois structure of nonabelian type arises from the action of a distinct Hopf algebra.

F-algebra isomorphisms
======================

In this section we investigate the $ F $-algebra structure of the Hopf algebras giving Hopf-Galois structures on $ L/F $. We assume that the characteristic of $F$ is not $ 2 $: this ensures the Hopf algebras are separable, hence semisimple, so that each has an Artin-Wedderburn decomposition (see section 3C of [@CUR1990]).

We fix some notation. Since $L/F$ is a quaternionic extension it has a 