Let $G$ be a finite group with the discrete topology. To set terminology:
a $G$-sphere is a sphere equipped with a continuous $G$-action
a $G$-representation sphere is a $G$-sphere obtained from an orthogonal $G$-representation $V$ by taking its unit sphere $S(V)\subset V$
From what I've heard, there exist $G$-spheres that are not homeomorphic (as $G$-spaces) to any $G$-representation sphere. However, I haven't been able to construct one, or find any reference for this; any help would be appreciated.