Let L/K be a primitive purely inseparable extension of fields of characteristic p, [L : K] >p, p odd. It is well known that L/K is Hopf Galois for some Hopf algebra H, and it is suspected that L/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. We provide some explicit examples of the dual, Hopf Galois, structure on L/K.