Chern Class and Stiefel-Whitney class for $U(N)$ and $PSU(N)$ vector bundle

Question

It is known that there is a relation between the first Chern class of $U(N)$ vector bundle and the Stiefel-Whitney class of the $PSU(N)$ vector bundle, for even $N$, i.e., $$c_1^{U(N)}= w_2^{PSU(N)}\mod N$$ where $w_2^{PSU(N)}$ is valued in $\mathbb{Z}_N$.

I am trying to know whether there is a relation between $w_4^{PSU(N)}$ and $c_2^{U(N)}$.

Any comments are appreciated!