In the stable homotopy category a map of spectra $f\colon X \rightarrow Y$ is called $\textit{phantom}$ is the induced map between the associated homology theories $X_* \rightarrow Y_*$ is zero, it is know that there are non-trivial phantom maps and these have been treated extensively. The most immediate reference I can think of is the paper "Phantom maps and homology theories" by Christensen and Strickland.

I was wondering if this was the case even in equivariant stable homotopy theory: i.e. suppose $G$ is a group, $f\colon X \rightarrow Y$ is a morphisms of $G$-spectra and for all the subgroups $H \leq G$ we have $\pi_*^H(f)\colon \pi_*^H(X)\rightarrow \pi_*^H(Y)$ is trivial. Do you have any reference for a study of this kind of maps? Do the results in the non-equivariant scenario generalize without any complication?

Also, it would be interesting to see what is their relationship with the maps $f$ such that $\Phi^H(f)$ is zero, where $\Phi^H$ indicates the geometric fixed points.