  We consider Hopf-Galois structures on separable (but not necessarily normal)
field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure
of $L/K$ then $G=\Gal(E/K)$ can be viewed as a permutation group of degree $n$.
We show that $G$ has derived length at most $4$, but that many permutation
groups of squarefree degree and of derived length $2$ cannot occur. We also
show that if $\gcd(n,\varphi(n))=1$ then $L/K$ admits at most one Hopf-Galois
structure. We then investigate in detail the case where $n=pq$ where $q \geq 3$
and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a
safeprime). We list the permutation groups $G$ which can arise, and we
enumerate the Hopf-Galois structures for each $G$. There are six such $G$ for
which the corresponding field extensions $L/K$ admit Hopf-Galois structures of
both possible types.
