Modular fixed points in equivariant homotopy theory

We show that the derived $\mathrm{\infty }$-category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the $\mathrm{\infty }$-category of equivariant spectra. On such module categories we define a modular fixed point functor using geometric fixed points followed by an extension of scalars and identify it with the modular fixed point functor on derived permutation modules introduced by Balmer-Gallauer. As an application, we show that the Picard group of such a module category for a $p$-group is given by the group of class functions satisfying the Borel-Smith conditions. In the language of representation theory, this result was first obtained by Miller.