For finite abelian G with Sylow p-subgroup N_p, the KU_G/p-local sphere equals homotopy fixed points of a p-complete KO_{N_p}-module and a wedge of equivariant Morava K-theory spheres, with computed Z-graded and RO(G)-graded homotopy Mackey functors.
[Mah70] Mark Mahowald
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RO(C2)-graded Hurewicz images of C2-equivariant Eilenberg-MacLane spectra are determined by the number of linearly independent vector fields on spheres.
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On the equivariant $KU_G$-local sphere for finite abelian groups
For finite abelian G with Sylow p-subgroup N_p, the KU_G/p-local sphere equals homotopy fixed points of a p-complete KO_{N_p}-module and a wedge of equivariant Morava K-theory spheres, with computed Z-graded and RO(G)-graded homotopy Mackey functors.
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A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres
RO(C2)-graded Hurewicz images of C2-equivariant Eilenberg-MacLane spectra are determined by the number of linearly independent vector fields on spheres.