The local Langlands correspondence of essentially unipotent supercuspidal representations for disconnected reductive groups

We construct the local Langlands correspondence of essentially unipotent supercuspidal representations under the framework of rigid inner forms and prove a certaion functoriality and compatibilities. This result is stronger than the analogous one in [Sol20], which did not care about rigidifications of inner twists. We also generalize this correspondence for disconnected reductive groups. We expect to use this result for extension of the explicit local Langlands correspondence in [Kal21] for more general supercuspidal representations.