Fixed Points of G-CW-complex with Prescribed Homotopy Type

For a group G of not prime power order, Oliver showed that the obstruction for a finite CW-complex F to be the fixed point set of a contractible finite G-CW complex is the Euler characteristic of F. We show that the similar problem for F to be fixed point set of a finite G-CW complex of some given homotopy type is still the Euler characteristic. We also give additional information on the obstruction in terms of the whole and the individual connected components of the fixed point set.