This thesis consists of three main parts, prefaced by a general introduction.The first part is based on a paper joint with Richard Wong. We exhibit an ∞-categorical decom-position of the stable module category of a general finite group, and we show that, in the case ofcertain particularly simple finite p-groups, this decomposition can be interpreted as an instance ofGalois descent. We then use this perspective to produce proof-of-concept calculations of the group ofendotrivial modules for these p-groups.In the second part, we move on to computations for more complicated groups. Of particularinterest will be the case of the extraspecial groups, which have traditionally played a fundamentalrole in the theory of endotrivial modules. We analyse the Picard spectral sequence for the extraspecialgroups and show that the E2-page inherits a great deal of structure from a certain Tits building ofisotropic subspaces with respect to a quadratic form.In the third and final part, we move on to study the Dade group of endopermutation modules. Weinvestigate how it can be realised as the Picard group of a certain ∞-category of genuine equivariantspectra. On our way, we produce a general framework for studying modules whose endomorphismsare trivial up to a specified subcategory of the representation category. This produces invariants thatinterpolate between the group of endotrivial modules and the Dade group, as well as other moreexotic invariants that are of independent interest.