Let $E/K$ be a finite Galois extension with cyclic abelian Galois group. Let $L/K$ be the unique subextension such that $[E:L]$ is coprime to $p$ and $[L:K]$ is a $p$-power. Let $\tau$ be a generator of $\mathrm{Gal}(L/K)$. Then there exists a lift $\tilde\tau\in \mathrm{Gal}(E/K)$ of $\tau$ such that $\tilde \tau$ has the same order as $\tau$ (that is, $[L:K]$).
Let $x\in E$, and assume that $\prod_{i=0}^{[L:K]-1}\tilde\tau^i(x)\in L$. Does it follow that $x\in L$?
No. Take $p > 2$, $K = \mathbf{Q}$, and $E = \mathbf{Q}(\zeta_{p^2})$. Let $x = \zeta_p$. Then $x$ lies in the fixed field of $\tilde{\tau}$, so
$$\prod_{i=0}^{p-1} \tilde{\tau}^i x = \prod_{i=1}^{p-1} \zeta_p = 1,$$
but $\zeta_p \notin L$.