Nilpotent and abelian Hopf-Galois structures on field extensions

Let $L/K$ be a finite Galois extension of fields with group $\Gamma$. When $\Gamma$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the Sylow subgroups of $\Gamma$. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When $\Gamma$ is abelian, we give conditions under which every abelian Hopf-Galois structure on $L/K$ has type $\Gamma$. We also give a criterion on $n$ such that \emph{every} Hopf-Galois structure on a cyclic extension of degree $n$ has cyclic type.