Eilenberg Mac Lane spectra as p-cyclonic Thom spectra

Hopkins and Mahowald gave a simple description of the mod $p$ Eilenberg Mac Lane spectrum $\mathbb{F}_p$ as the free $\mathbb{E}_2$-algebra with an equivalence of $p$ and $0$. We show for each faithful $2$-dimensional representation $\lambda$ of a $p$-group $G$ that the $G$-equivariant Eilenberg Mac Lane spectrum $\underline{\mathbb{F}}_p$ is the free $\mathbb{E}_{\lambda}$-algebra with an equivalence of $p$ and $0$. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral $2$-subgroups of O(2). The main new idea is that $\underline{\mathbb{F}}_p$ has a simple description as a $p$-cyclonic module over $\rm{THH}(\mathbb{F}_p)$. We show our result is the best possible one in that it gives all groups $G$ and representations $V$ such that $\underline{\mathbb{F}}_p$ is the free $\mathbb{E}_V$-algebra with an equivalence of $p$ and $0$.