Local Langlands correspondence for covering groups of tori, and the packet-indexing groups

We determine the finite group $\mathcal{S}$ parametrizing a packet in the local Langlands correspondence for a Brylinski-Deligne covering group of an algebraic torus, under some assumption on ramification. Especially, this work generalizes Weissman’s result on covering groups of tori that split over an unramified extension of the base field.