On the equivariant $K{U}_G$-local sphere for finite abelian groups

Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $K{U}_G\mathrm{/}p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $K{O}_{N_p}$-module spectrum. Then we compute the $\mathbb{Z}$-graded homotopy Mackey functors of the $K{U}_G$-local sphere spectrum. This result generalizes the computation of arXiv:2303.12271 for finite $p$-groups, where $p$ is an odd prime. Finally, by comparing the Bousfield classes of $K{U}_G\mathrm{/}p$ and $G$-equivariant Morava $K$-theory, we prove that the $K{U}_G\mathrm{/}p$-local sphere spectrum is equivalent to a wedge sum of equivariant Morava $K$-theory localized sphere spectra, and describe the $RO(G)$-graded homotopy Mackey functors of the $K{U}_G\mathrm{/}p$-local sphere spectrum.