Cohomology groups $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$

Question

I'm trying to compute $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$. Here the semi-direct product is given by the outer automorphism of $PSU(4)$.

By Serre spectral sequence, we have $$H^p(B\mathbb{Z}_2,H^q(BPSU(4),\mathbb{Z}_2))\Rightarrow H^{p+q}(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2).$$

The relevant piece is

There could be several differentials, I'm not sure which of them actually exist.

Thank you!