I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (should it actually be a compact Lie group?) in my question. Also, for a group $G$ the corresponding stable homotopy category $SH_G$ (should I use another notation?) also depends on the choice of a universe; does one get a canonical category taking a complete universe?

As far as I understand equivariant stable homotopy theory, in $SH_G$ one should take spectra $\Sigma_G (G/H)_+$ corresponding to all homogenious spaces $G/H$ to obtain a family of compact generators. Now, I want to know something on morphisms between these generators. In particular, is the group $Mor_{SH_G} (\Sigma_G (G/H_1)_+, \Sigma_G (G/H_2)_+[i])$ zero for any $i>0$ and all subgroups of $G$? This question seems to be related to A heart for stable equivariant homotopy theory Is there any text where I can find results of this sort? Possibly, one can say something about these morphism groups (say) using duality and apply some well-known results after that?

Any hints would be very welcome!