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

Let $L/K$ be any finite separable extension with normal closure $\widetilde{L}/K$. An extension $L'/K$ is said to be \textit{parallel to L/K} if $L'$ is an intermediate field of $\widetilde{L}/K$ with $[L':K]=[L:K]$. We study the following question -- Given that $L/K$ admits a Hopf--Galois structure of type $N$, does it imply that every extension parallel to $L/K$ 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.