Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group $G$, we compute a certain subgroup $IO'(G)$ of the representation ring $RO(G)$. This allows us to prove that a finite perfect group $G$ has a smooth $2$-proper action on a sphere with isolated fixed points at which the tangent representations of $G$ are mutually nonisomorphic if and only if $G$ contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer $n \ge 1$ and primes $p, q$, we prove similar results for the group $G = A_{n}$, $\operatorname{SL} _{2}(\mathbb{F} _{p})$, or ${\operatorname{PSL}} _{2}(\mathbb{F} _{q})$. In particular, $G$ has Smith equivalent representations that are not isomorphic if and only if $n \ge 8$, $p \ge 5$, $q \ge 19$.