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For a finite group $G$ not of prime power order, Oliver (1996) has answeredthe question which manifolds occur as the fixed point sets of smooth actions of$G$ on disks (resp., Euclidean spaces). We extend Oliver's result to compact(resp., open) smooth $G$-manifolds $M$ pseudo-equivalent to $Y$, a finite$\mathbb{Z}$-acyclic $G$-CW complex such that the fixed point set $Y^G$ isnon-empty, connected, and $\chi(Y^G) \equiv 1 \pmod{n_G}$, where $n_G$ is theOliver number of $G$. We prove that the answer to the question above does notdepend on the choice of $Y$.  For a finite connected $G$-CW complex $Y$ such that $Y^G$ is non-empty andconnected, called a $G$-template, we prove that a compact stably parallelizablemanifold $F$ occurs as the fixed point set $M^G$ of a compact smooth$G$-manifold $M$ pseudo-equivalent to $Y$, if and only if $\chi(F) \equiv\chi(Y^G) \pmod{n_G}$. Moreover, there exists a compact smooth fixed point free$G$-manifold pseudo-equivalent to a $G$-template $Y$, if and only if $\chi(Y^G)\equiv 0 \pmod{n_G}$. In particular, similarly as for actions on disks, thereexists a compact smooth fixed point free $G$-manifold pseudo-equivalent to thereal projective space $\mathbb{R}{\rm P}^{2n}$ for an integer $n \geq 1$, ifand only if $G$ is an Oliver group. Finally, we prove that each finite Olivergroup $G$ has a smooth fixed point free action on $\mathbb{R}{\rm P}^{2n}$itself for some integer $n \geq 1$.

Fixed point sets of smooth $G$-manifolds pseudo-equivalent to a $G$-template