Local coefficients and Gelfand–Graev representations for non-split covers on $\mathrm{SL}(2)$

A purely local approach has been developed by Krishnamurthy and Kutzko to compute the Langlands–Shahidi local coefficients for ${\rm SL}(2)$ via types and covers à la Bushnell–Kutzko. In this paper, we extend their method to the non-split case and complete their project. We also study the algebraic structure of Gelfand–Graev representations, which generalizes the results of Chan–Savin and Mishra–Pattanayak to ${\rm SL}(2)$ over non-archimedean local fields without any restriction on the characteristic.