first named author acknowledges funding support from the Faculty of Natural Sciences at Keele University. We are grateful to Prof. Alan Koch for his comments on an early draft of this paper, and to the anonymous referee for improvements to the exposition and interpretation of our results.

Structures on the extension
===========================

Let $L/F$ be a Galois extension of fields with Galois group $G$ isomorphic to the quaternion group of order $8$. Let $G$ have generators $\sigma$ and $\tau$, that is $$G=\langle\sigma,\tau|\sigma^4=\tau^4=1,\sigma^2=\tau^2, \sigma\tau=\tau\sigma^{-1}\rangle.$$

There are 5 isomorphism types of groups of order 8: the elementary abelian group $C_2\times C_2\times C_2$, $C_4\times C_2$, the cyclic group $C_8$, the dihedral group $D_4$ and the quaternion group $Q_8$. As mentioned in the introduction, [@BYO2018 Table A.1] includes a count of the Hopf-Galois structures admitted by $ L/F $, which we reproduce in below. The same count appears in work of Crespo and Salguero [@CRE2017 Table 3], as an application of an algorithm written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree.

             Type              Number of structures
  --------------------------- ----------------------
   $C_2\times C_2\times C_2$            2
        $C_4\times C_2$                 6
             $C_8$                      6
             $Q_8$                      2
             $D_4$                      6

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