  We construct Steinberg summands of $G$-equivariant spectra with
$\mathrm{GL}_n(\mathbb{F}_p)$-action. We prove a lemma about their fixed points
when $G$ is a $p$-group, and then use this lemma to compute the fixed points of
the Steinberg summand of the equivariant classifying space of
$(\mathbb{Z}/p)^n$. These results will be used in a companion paper to study
the layers in the mod $p$ symmetric power filtration for
$H\underline{\mathbb{F}}_p$.
