Liftability of periodic mapping classes under alternating covers

Let $S_g$ be the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. Let $A_n$ denote the alternating group on $n$ letters. We derive necessary and sufficient conditions under which a periodic mapping class has a conjugate that lifts under the branched cover $S_g \to S_g/A_n$ induced by an action of $A_n$ on $S_g$. This provides a classification of the subgroups of $\mathrm{Mod}(S_g)$ that are isomorphic to $A_n \rtimes \mathbb{Z}_m$, up to a certain equivalence that we call weak conjugacy. As an application, we show that for $n \geq 7$, such a subgroup of $\mathrm{Mod}(S_g)$ cannot have an irreducible periodic mapping class. Furthermore, we show that for $n \geq 5$ and $n \neq 6$, if the order of such a subgroup is greater than $5g-5$, then $m \leq 26$. Moreover, for $g \geq 2$ and $n \geq 5$, we establish that there exists no subgroup of $\mathrm{Mod}(S_g)$ that is isomorphic to $A_n \rtimes \mathbb{Z}$, where the $\mathbb{Z}$-component is generated by a power of a Dehn twist. Finally, we provide a complete classification of the weak conjugacy classes of such subgroups in $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$.