The $\mathbb{Z}\mathrm{/}p$-equivariant cohomology of genus zero Deligne-Mumford space with $1+p$ marked points

We prove that the Serre spectral sequence of the fibration ${\bar{\mathcal{M}}}_{0,1+p}\to E\mathbb{Z}\mathrm{/}p{\times }_{\mathbb{Z}\mathrm{/}p}{\bar{\mathcal{M}}}_{0,1+p}\to B\mathbb{Z}\mathrm{/}p$ collapses at the $E_2$ page. We use this to prove that: for any element of the $\mathbb{Z}\mathrm{/}p$-equivariant cohomology with ${\mathbb{F}}_p$-coefficients of genus zero Deligne-Mumford space with $1+p$ marked points, if this element is torsion then it is non-equivariant. This concludes that the only “interesting” $\mathbb{Z}\mathrm{/}p$-equivariant operations are quantum Steenrod power operations.