Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension $L\mathrm{/}K$ of degree $p^n$, for $p$ an odd prime number, $n\ge 3$. For $p>n$, we prove that $L\mathrm{/}K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group $N$ of order $p^n$, with commutator subgroup of order $p$, we prove that if $L\mathrm{/}K$ has a Hopf Galois structure of type $N$, then it has a Hopf Galois structure of type $A$, where $A$ is an abelian group of order $p^n$ and having the same number of elements of order $p^m$ as $N$, for $1\le m\le n$.