Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

For G any finite group, the left and right regular representations λ, respectively ρ of G into Perm(G) map G into InHol(G)=ρ(G)⋅Inn(G). We determine regular embeddings of G into InHol(G) modulo equivalence by conjugation in Hol(G) by automorphisms of G, for groups G that are semidirect products G=Zh⋊Zk of cyclic groups and have trivial centers. If h is the power of an odd prime p, then the number of equivalence classes of regular embeddings of G into InHol(G) is equal to twice the number of fixed-point free endomorphisms of G, and we determine that number. Each equivalence class of regular embeddings determines a Hopf Galois structure on a Galois extension of fields L/K with Galois group G. We show that if H1 is the Hopf algebra that gives the standard non-classical Hopf Galois structure on L/K, then H1 gives a different Hopf Galois structure on L/K for each fixed-point free endomorphism of G.