On lifting representations and actions on curves of the metacyclic groups $C_{p^s}\rtimes C_m$

For a prime $p$, a pair $(s,m)\in\mathbb{N}^2$ with $m$ relatively prime to $p$, a homomorphism $χ:C_m\rightarrow\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\rtimes_χ C_m$, denote its $p$-Sylow subgroup $C_{p