Several types of solvable groups as automorphism groups of compact Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order $p^m q^n$. Moreover, we show that Zomorrodian's bound for $p$-groups $G$ with $p\geq 5$, namely $|G|\leq \frac{2p}{p-3}(g-1)$, actually holds for any group $G$ for which $p\geq 5$ is the smallest prime divisor of $|G|$.