Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

Let $L\mathrm{/}K$ be any finite separable extension with normal closure $\tilde{L}\mathrm{/}K$. An extension $L^{\mathrm{\prime }}\mathrm{/}K$ is said to be \textit{parallel toL/K} if $L^{\mathrm{\prime }}$ is an intermediate field of $\tilde{L}\mathrm{/}K$ with $[L^{\mathrm{\prime }}:K]=[L:K]$. We study the following question – Given that $L\mathrm{/}K$ admits a Hopf–Galois structure of type $N$, does it imply that every extension parallel to $L\mathrm{/}K$ also admits a Hopf–Galois structure of type $N$? We completely solve this problem when the degree $[L:K]$ is a prime power and the type $N$ is cyclic. Our approach is group-theoretic and uses the work of Greither–Pareigis and Byott.