Double covers of tori for p=2 and genuine functions

In his work on $L$-embeddings and double covers of tori over local fields, Kaletha associates to a maximal torus $S$ in a connected reductive group a canonical double cover $S(F{)}_{\pm }$ and introduces a genuine sign function $a_S$ associated to a genuine character of $S(F{)}_{\pm }$. We give two constructions of this genuine sign function over a non-archimedean local field, including residual characteristic $2$. The first uses rootwise lifts to the double cover; the second, in characteristic zero, uses the exponential map. Under explicit ramification conditions, both constructions yield genuine sign functions which, in odd residual characteristic, agree with his construction. We also illustrate the case of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$. A brief announcement of the first construction appears in Appendix D of that work.