We will describe two combinatorial problems inspired by finite automorphism groups of compact Riemann surfaces of genus two or greater: enumerating the topological actions of a finite group on surfaces and determining the set of genera of surfaces admitting such a group action, called the genus spectrum. We will illustrate results in the important case of quasiplatonic cyclic group actions. Specifically, using formulas of Benim and Wootton (2014), we show that the number of quasiplatonic cyclic group actions includes the number of regular dessins d'enfants with a cyclic group of symmetries on these surfaces. In addition, by optimizing the Riemann-Hurwitz formula under certain conditions, we discuss 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).