The author classifies finite groups acting on smooth plane curves of degree at least four. Furthermore, he gives an upper bound for the order of automorphism groups of smooth plane curves and determines the exceptional cases in terms of defining equations. This paper also contains a simple proof of the uniqueness of smooth plane curves with the full automorphism group of maximum order for each degree.
"Automorphism groups of smooth plane curves." Kodai Math. J. 42 (2) 308 - 331, June 2019. https://doi.org/10.2996/kmj/1562032832