Question: Let $G$ be a compact Lie group (you can assume that $G$ is finite if you like). Is the category of finite $G$-spectra idempotent complete?
Here, by "finite $G$-spectra", I mean those objects in the category of genuine $G$-spectra which can be constructed via a finite number of sums and cofiber sequences from orbit cells $S^n \wedge (G/H)_+$ where $H \subseteq G$ is a closed subgroup. This includes all representation spheres. By "category", I mean either the stable $\infty$-category or the triangulated category, whichever you prefer (in the stable case, a category is idempotent complete iff its homotopy category is, so it doesn't affect the question.)
When $G$ is trivial, the answer is yes, but I suspect the answer in general is no, -- maybe already for $G = C_2$?
By a theorem of Thomason, triangulated subcategories of a triangulated category $\mathcal T$ which generate $\mathcal T$ under splitting of idempotents are in bijection with subgroups of $K_0(\mathcal T)$. So another way to phrase the question is the following: let $\mathcal T$ be the category of compact $G$-spectra. Then is t$K_0(\mathcal T)$ generated as a group by the classes of $G$-orbits $\Sigma^n(G/H)_+$?