It is well known that a group of automorphisms $G$ of an unbordered Klein surface $X$ of topological genus $g\geq 2$ in the orientable case and $g\geq 3$ otherwise has at most $84(g-\varepsilon)$ elements, where $\varepsilon =1$ or $2$ respectively. In the middle of the fifties, Oikawa used the cardinality $k$ of a $G$-invariant subset to introduce the bound $\lvert G\rvert \leq 12(g-1)+6k$ in the orientable case. Much later, T. Arakawa has generalized this result, involving $s=2$ or $3$ such subsets and showing in addition that the bound for $s=3$ is sharp for infinitely many configurations. Here we improve the bound of Arakawa for $s=2$, showing in particular that the last is never attained. In both orientable and non-orientable case, we also find bounds for arbitrary $s$ and show their sharpness for infinitely many topological configurations. Using another well known theorem of Oikawa and the canonical Riemann double cover, we get similar results for bordered Klein surfaces.
"On automorphisms of Klein surfaces with invariant subsets." Osaka J. Math. 50 (1) 251 - 269, March 2013.