Equivariant homotopy theory focuses on spaces together with some group action on them.  Jeroen van der Meer and Richard Wong have [a paper](https://arxiv.org/abs/2107.06308) where they use equivariant methods to compute the Picard group of the stable module category of representations for certain finite groups.  I was wondering if there are more results of a similar flavor, providing applications of equivariant homotopy theory to representation theory (or I suppose group theory in general).