Given any finite groups \(\Gamma\) and \(G\) of the same order, let \(e(\Gamma,G)\) denote the number of Hopf-Galois structures of type \(G\) on any Galois \(\Gamma\)-extension. By work of [C. Greither and B. Pareigis, J. Algebra 106, 239–258 (1987; Zbl 0615.12026)] and [N. P. Byott, Commun. Algebra 24, 3217–3228 (1996; Zbl 0878.12001)], it is known that \[ e(\Gamma,G) = \frac{|\mathrm{Aut}(\Gamma)|}{|\mathrm{Aut}(G)|}\cdot\#\{\mbox{regular subgroups of \(\mathrm{Hol}(G)\) isomorphic to \(\Gamma\)}\}, \] where \(\mathrm{Hol}(G)\) denotes the holomorph of \(G\). In the paper under review, assuming that \(n\) is an odd number whose radical is a Burnside number, the authors use this formula to compute \(e(\Gamma,G)\) when \(\Gamma\) and \(G\) are both of the shape \(\mathbb{Z}_n\rtimes \mathbb{Z}_2\). More specifically, any such group is isomorphic to a direct product of a cyclic group \(C_l\) and a dihedral group \(D_{2k}\), with \(\gcd(l,k)=1\) and \(n = lk\). The main result (Theorem 1.3) states that for \(\Gamma = C_{l_1}\times D_{2k_1}\) and \(G = C_{l_2}\times D_{2k_2}\) with \(\gcd(l_1,k_1)=\gcd(l_2,k_2)=1\) and \(n = l_1k_1= l_2k_2\), one has \[ e(\Gamma,G) = \frac{l_1l_2}{\gcd(l_1,l_2)\mathfrak{R}(l_1)}\cdot 2^{|\pi(k_2)|}, \] where \(\mathfrak{R}(l_1)\) denotes the radical of \(l_1\) and \(\pi(k_2)\) denotes the set of prime factors of \(k_2\). The proof involves a careful analysis of regular embeddings of \(\Gamma\) into \(\mathrm{Hol}(G)\), which reduces to counting solutions to certain congruence equations. When \(\Gamma = G = C_l\) or \(\Gamma = G = D_{2k}\), the count for \(e(\Gamma,G)\) is known by [N. P. Byott, J. Algebra 381, 131–139 (2013; Zbl 1345.12002)] and [T. Kohl, J. Algebra 542, 93–115 (2020; Zbl 1462.12002)], respectively; unlike the paper under review \(l,k\) can be any natural numbers in these two papers. The authors remark that in these special cases their counts for \(e(\Gamma,G)\) agree with the previously known results.