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 $\chi:C_m\rightarrow\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\rtimes_{\chi} C_m$, denote its $p$-Sylow subgroup $C_{p^s}$ by $H$, and consider a $k[G]$-module $V$. Let $R$ be a complete discrete valuation ring of residue field $k$ and mixed characteristic $(0,p)$ that contains a primitive $p^s$-th root of unity. If $\chi$ is injective, we present two necessary and sufficient criteria for lifting $V$ to an $R[G]$-module $\widetilde{V}$ which is a free $R$-module: (i) when no extra requirement is made on $\widetilde{V}$ and (ii) when we require $\widetilde{V}^{C_{p^s}}=\{0\}$. The criteria correct several results in the literature and we use them to prove that, if $\chi$ is injective and $G$ acts faithfully on a connected smooth projective curve $X$ over $k$, then, under mild hypotheses satisfied if $X\rightarrow X/G$ is a Harbater--Katz--Gabber cover, the $k[G]$-module $H^0(X,\Omega_X)$ has a lift $\widetilde{V}$ to $R$ with $\widetilde{V}^{C_{p^s}}=\{0\}$. With $B$ as the field of fractions of $R$, we prove the following obstruction when $p$ is odd, $G/\operatorname{Ker}(\chi)$ has even order, and $X/C_{p^s}\cong\mathbb{P}^1_k$: if no such lift $\widetilde{V}$ exists with the $B[H]$-module $\widetilde{V}\otimes_R B$ defined over $\mathbb{Q}$, then the action of $G$ on $X$ does not lift to $R$.