Lifting semisimple characters of $p$-adic types from fixed-point subgroups

Given a $p$-adic group $G=\mathbf{G}(F)$ and a finite group $\Gamma\subset\mathrm{Aut}_F(\mathbf{G})$ such that the fixed-point subgroup $\mathbf{G}^\Gamma$ is reductive, we show that every semisimple character (in the sense of Bushnell and Kutzko) of a type for $G^\Gamma = \mathbf{G}^\Gamma(F)$ arises as the restriction of a semisimple character of a type for $G$. We achieve this by explicitly lifting the truncated Kim--Yu datum (or character-datum) that parametrizes the semisimple character for $G^\Gamma$ to a character-datum that parametrizes a semisimple character for $G$. Our proof, which is of independent interest, uses state-of-the-art techniques and, as a special case, defines a lift of a Howe factorization of a character of a maximal torus of $G^\Gamma$.