In 1973, Macbeath found a general formula for the number of points fixed by an arbitrary orientation preserving automorphism of a Riemann surface $X$. It was given in terms of a group $G$ of conformal automorphisms of $X$ and the ramification data of the covering $X\to X/G$, which corresponds to the so called universal covering transformation group. In these terms, for the case of a cyclic group of automorphisms of an unbordered non-orientable Klein surface, the formula was given later by Izquierdo and Singerman and here we find formulas valid for an arbitrary (finite) group $G$ of automorphisms.
G. Gromadzk. "On fixed points of automorphisms of non-orientable unbordered Klein surfaces." Publ. Mat. 53 (1) 73 - 82, 2009.