Hopf Galois Structures on Primitive Purely Inseparable Extensions

Let $L\mathrm{/}K$ be a primitive purely inseparable extension of fields of characteristic $p$, $[L:K]>p\mathrm{.}$ It is well known that $L\mathrm{/}K$ is Hopf Galois for some Hopf algebra $H$, and it is suspected that $L\mathrm{/}K$ is Hopf Galois for numerous choices of $H$. We construct a family of $K$-Hopf algebras $H$ for which $L$ is an $H$-Galois object. For some choices of $K$ we will exhibit an infinite number of such $H\mathrm{.}$ We provide some explicit examples of the dual, Hopf Galois, structure on $L\mathrm{/}K\mathrm{.}$