On the $p$-primary subgroups of the cohomology of the classifying space of $PU_n$

Let $PU_n$ denote the projective unitary group of rank $n$, and let $BPU_n$ be its classifying space. We extend our previous results to a description of $H^s(BPU_n;\mathbb {Z})_{(p)}$ for $s \lt 2p+9$ by showing that the $p$-primary subgroup of $H^s(BPU_n;\mathbb {Z})$ is isomorphic to $\mathbb {Z}/p$ for $s=2p+5$ and is trivial for $s = 2p+7$ and $s = 2p+8$, where $p$ is an odd prime which divides $n$.