@DenisNardin From that paper, what I could understand is that there is a mysterious part of Picard group of G-spectra from Picard group of Burnside ring via Brown representability. This seems to be the interesting part, because none of the S^V are there for C_p^k, but I have no idea how to construct explicit example in this part. Do you have any V-suspension of a G-CW complex as an example of this flavor, say for G= C_2 or C_4?

@MingcongZeng The only way I know how to construct those elements are as retracts of wedges of the sphere spectrum. Probably you can wrestle a concrete description out of that, by picking a geometric representative of the idempotent that you are splitting off, but I have never seen it done

@DenisNardin Could you elaborate more on constructing them as retracts of wedges of sphere?

lol