The moduli space M‾0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action on the cohomology of M‾0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M‾0,n, equivariant with respect to the symmetric group action. Moreover we prove that H2k(M‾0,n) for k≤3 and H2k(M‾0,n)⊕H2k−2(M‾0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P1n of the configuration space of n points on P1 and more generally on the cohomology of the moduli space M‾0,n(Pm−1,1) of stable maps.