Let $G$ be a finite group and $X$ an orthogonal $\mathbb{Z}[\frac{1}{|G|}]$-local spectrum with an $BK$-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 $\mathbb{Z}[\frac{1}{|G|}]$-local equivalence.