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For a group $G$ of not prime power order, Oliver showed that the obstructionfor a finite CW-complex $F$ to be the fixed point set of a contractible finite$G$-CW-complex is the Euler characteristic $\chi(F)$. He also has the similarresults for compact Lie group actions. We show that the analogous problem for$F$ to be the fixed point set of a finite $G$-CW-complex of some given homotopytype is still determined by the Euler characteristic. Using trace maps in$K_0$, we also see that there are interesting roles for the fundamental groupand the component structure of the fixed point set.

Fixed Point Sets and the Fundamental Group II: Euler Characteristics