 Solution

Under the intended convention that the finite group is nontrivial, take ${\mathbb{CP}}^2$. It is simply connected by its CW complex structure. The Lefschetz fixed-point theorem says that a continuous self-map of a compact triangulable space with nonzero rational Lefschetz number has a fixed point.

For any continuous $f:{\mathbb{CP}}^2\to {\mathbb{CP}}^2$, write $f^{\ast }x=dx$ for the integral degree-two generator, with $d\in \mathbb{Z}$. Preservation of the cup product gives $f^{\ast }(x^2)=d^2{x}^2$. Only degrees zero, two and four contribute to the rational Lefschetz number, soConsequently every such map has a fixed point. In a free group action, each nonidentity group element must act by a fixed-point-free homeomorphism. This is impossible here, so no nontrivial finite group can act freely. This is the fixed-point property of even-dimensional complex projective space for complex dimension two. If the trivial group is included, the literal request has no example, since the trivial group acts freely on every space.