by $j(l\otimes h)(x)=lh(x)$ for $l,x\in L$, $h\in H$, is bijective.

We note that in this definition $ L/F $ may be taken to be merely an extension of commutative rings. However, in this paper we will be concerned exclusively with fields, specifically the case where $L/F$ is Galois (in the usual sense).

Since a Hopf-Galois structure on an extension $ L/F $ consists of a Hopf algebra $ H $ and an action of $ H $ on $ L $, it is possible for distinct Hopf-Galois structures on $ L/F $ to involve Hopf algebras that are isomorphic, either as $ F $-Hopf algebras or as $ F $-algebras. These phenomena have recently been studied in papers such as [@TARP2017] and [@TARP2017_2]. In particular, [@TARP2017_2] studies in detail the Hopf-Galois structures admitted by a dihedral extension of fields of degree $ 2p $, where $ p $ is an odd prime. In this paper we perform a similar analysis of the Hopf-Galois structures admitted by a Galois extension of fields with Galois group isomorphic to $ Q_{8} $, the quaternion group of order $ 8 $. We call such extensions *quaternionic*. In addition to continuing and complementing the work 