  We study the inertia stack of [M_{0,n}/S_n], the quotient stack of the moduli
space of smooth genus 0 curves with n marked points via the action of the
symmetric group S_n. Then we see how from this analysis we can obtain a
description of the inertia stack of H_g, the moduli stack of hyperelliptic
curves of genus g. From this, we can compute additively the Chen-Ruan (or
orbifold) cohomology of H_g.
