Steinberg Summands in the free $\mathbb{F}_p$-module on the Equivariant Sphere Spectrum

Let $G$ be a finite $p$-group. The Eilenberg-Maclane spectrum of the constant Mackey functor $\underline{\mathbb{F}}_p$, denoted $H\underline{\mathbb{F}}_p$, is modeled by the free $\mathbb{F}_p$-module on the $G$-equivariant sphere spectrum. With this construction, one has a `word length' filtration $\{(H\underline{\mathbb{F}}_p)_n\}_{n\ge 1}$. Our main theorem is that the $k$-th layer $(H\underline{\mathbb{F}}_p)_{p^k}/(H\underline{\mathbb{F}}_p)_{p^{k-1}}$ is $p$-locally equivalent to the $k$-fold suspension of the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}/p)^k$. This is a generalization of the main result of [21]. We also show that when one smashes this filtration with $H\underline{\mathbb{F}}_p$, the filtration splits into its layers. The future goal of this work is to compute the $C_p$-equivariant dual Steenrod algebra $H\underline{\mathbb{F}}_p\wedge H\underline{\mathbb{F}}_p$ when $p>2$, via explicit cellular constructions of equivariant classifying spaces.