We describe two combinatorial problems in the theory of automorphism groups of compact Riemann surfaces of genus two or greater: enumerate the topological actions of a finite group on surfaces and determine the set of genera of surfaces admitting such a group action, called the genus spectrum. We illustrate results in the important case of quasiplatonic cyclic group actions. In Chapter 3, we use formulas of Benim and Wootton (2014) to derive a closed formula for the number of quasiplatonic topological actions of the cyclic group. The formula implies that the number of quasiplatonic cyclic group actions is roughly one-sixth the number of regular dessins d'enfants with a cyclic group of symmetries. By optimizing the Riemann-Hurwitz formula under certain conditions, we discuss in Chapter 4 the second-smallest genus of a surface admitting a quasiplatonic cyclic group action, which appears to resemble the known minimal genus action due to Harvey (1966). We then determine a closed form for the second-smallest genus when the smallest prime divisor of a positive integer has exponent one.