@MarcHoyois This looks very interesting! I am thinking about similar things for a few weeks, and it makes what I am wondering kind of positive

Thank you!

@MingcongZeng This is probably not the D_4-spectrum you expect though, since one of the reflections in D_4 acts trivially...

Sorry that was a mistake. The underlying spectrum with D_4-action is actually the one you expect, so I think this D_4-spectrum should be exactly what you want.

In general if E is a G-spectrum and H is a group of order n, then the n-fold smash power of E is canonically a G≀H-spectrum: first pull back along a projection BG^n → BG and then norm along BG^n → (BG^n)/H = B(G≀H).