Let $G$ be a finite group and $X$ an orthogonal $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local spectrum with an $G$-action that is trivial on $\pi_*X$.

I want to show that then the map $X^{hG} \to X$ from the homotopy fixed points is a $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local equivalence.