  In this second part we prove that, if $G$ is one of the groups
$\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13
\pmod{24}$, then the fundamental group of every acyclic $2$-dimensional, fixed
point free and finite $G$-complex admits a nontrivial representation in a
unitary group $\mathrm{U}(m)$. 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.
