Endoscopic lifting of simple supercuspidal representations of SO2n+1 TO GL2n

We compute the characters of simple supercuspidal representations of the twisted ${\rm GL}_{2n}(F)$ and ${\rm SO}_{2n+1}(F)$ for a $p$-adic field $F$. Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of ${\rm SO}_{2n+1}(F)$ to ${\rm GL}_{2n}(F)$, under the assumption that $p$ is not equal to $2$.