Hopf-Galois structures of cyclic type on a dihedral or quaternionic extension

Question

Let $L/K$ be a dihedral or quaternionic finite field extension, that is such that $Gal(L/K)$ is either a dihedral or a quaternion group. How many Hopf-Galois structures of cyclic type are there on such extension? I am interested, in particular, when Gal(L/K) is isomorphic to either $D_{2^n}$ (or $Q_{2^n}$). I only know that that number is 6 when $Gal(L/K)\cong Q_8$ (source: https://arxiv.org/abs/1809.09497). I have conjectured that, in general, for all such cases, the number I am looking for is $2^{n-2}$.