In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_{2}(q)$ with $q>5$ and $q\equiv 5\ (\mathrm{mod}\ 24)$ or $q\equiv 13 \ (\mathrm{mod}\ 24)$ , then the fundamental group of every acyclic 2-dimensional, fixed point free and finite $G$ -complex admits a nontrivial representation in a unitary group $\mathbf{U}(m)$ . This completes the proof of the following result: every action of a finite group on a finite and contractible 2-complex has a fixed point.
