  Let $G$ and $N$ be two finite groups of the same order. It is well-known that
the existences of the following are equivalent: (a) a Hopf-Galois structure of
type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$
and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the
holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above
exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the
groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a
cyclic group instead, we shall give a complete characterization of the groups
$N$ for which $(G,N)$ is realizable.
