  The moduli space $\overline{\mathcal{M}}_{0,n}$ of $n$ pointed stable curves
of genus $0$ admits an action of the symmetric group $S_n$ by permuting the
marked points. We provide a closed formula for the character of the
$S_n$-action on the cohomology of $\overline{\mathcal{M}}_{0,n}$. This is
achieved by studying wall crossings of the moduli spaces of quasimaps which
provide us with a new inductive construction of $\overline{\mathcal{M}}_{0,n}$,
equivariant with respect to the symmetric group action. Moreover we prove that
$H^{2k}(\overline{\mathcal{M}}_{0,n})$ for $k\le 3$ and
$H^{2k}(\overline{\mathcal{M}}_{0,n})\oplus
H^{2k-2}(\overline{\mathcal{M}}_{0,n})$ 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 $S_n$-representation on the
cohomology of the Fulton-MacPherson compactification $\mathbb{P}^1[n]$ of the
configuration space of $n$ points on $\mathbb{P}^1$ and more generally on the
cohomology of the moduli space
$\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{m-1},1)$ of stable maps.
