We present first results concerning asymptotic (long-time) chaotic superdiffusion of the average kinetic energy of N Floquet (periodically kicked) particles globally interacting with each other. This superdiffusion is due to accelerator modes (accelerating periodic orbits) that are linearly stable in all the 2N directions of phase space (full stability). We exactly determine the phase-space locations and linear-stability characteristics of particular period-1 accelerator modes for arbitrary N. While the locations of these modes are independent of the interaction strength, their linear-stability characteristics depend on it. Unlike the case of a single particle (N=1), the 2N-dimensional "stability island" formed by a  fully stable accelerator mode is not impenetrable by chaos, due to Arnol'd diffusion. Despite this, we numerically observe a long-time superdiffusion in the presence of the fully stable accelerator modes above for small interaction strength. However, by gradually increasing the interaction at fixed kicking parameter, the accelerator modes become linearly unstable in some phase-space directions. Then, the superdiffusion turns into normal diffusion. The dependence of the anomalous superdiffusion exponent on several parameters is also numerically studied.