  For each $n\ge5$, we give an $S_n$-equivariant basis for
$H_4(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$, as well as for
$H_{2(n-5)}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$. Such a basis exists for
$H_2(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$ and for
$H_{2(n-4)}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$, but it is not known
whether one exists for $H_{2k}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$ when
$3\le k\le n-6$.
