On endosplit $p$-permutation resolutions and Brou\'{e}'s conjecture for $p$-solvable groups

Endosplit $p$-permutation resolutions play an instrumental role in verifying Brou\'{e}'s abelian defect group conjecture in numerous cases. We give a new characterization of all endosplit $p$-permutation resolutions and reduce the question of Galois descent of an endosplit $p$-permutation resolution to the Galois descent of the module it resolves. This is shown using techniques from the study of endotrivial complexes, the invertible objects of the bounded homotopy category of $p$-permutation modules. As an application, we show that a refinement of Brou\'{e}'s conjecture proposed by Kessar--Linckelmann holds for certain blocks of groups $G$ satisfying $G = O_{p',p,p'}(G)$ with abelian Sylow $p$-subgroup, the key reduction step in Harris--Linckelmann's verification of Brou\'e's conjecture for all $p$-solvable groups.