Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ where $\mathbb{Z}_2$ acts on $PSU(4)$ by the outer automorphism.
Since there is a (right split) short exact sequence of groups: $$1\to PSU(4)\to \mathbb{Z}_2\ltimes PSU(4)\to\mathbb{Z}_2\to1,$$ we have a fiber sequence $$BPSU(4)\to B(\mathbb{Z}_2\ltimes PSU(4))\to B\mathbb{Z}_2.$$ Inspired by my previous question, there seems to be no differentials in the Serre spectral sequence $$H^p(B\mathbb{Z}_2,H^q(BPSU(4),\mathbb{Z}_2))\Rightarrow H^{p+q}(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$$ for $p+q\le3$.
So I think $P$ can be identified with a principal $\mathbb{Z}_2\times PSU(4)$-bundle over $M$ whose isomorphism class is represented by the homotopy class of a pair of maps:
$$g:M\to B\mathbb{Z}_2$$ and $$h:M\to BPSU(4).$$
By my previous another question, for 3-manifold $M$, $h$ is equivalent to a map $$h':M\to B^2\mathbb{Z}_4.$$
Now the homotopy class of $g$ is a cohomology class in $H^1(M,\mathbb{Z}_2)$, the homotopy class of $h'$ is a cohomology class in $H^2(M,\mathbb{Z}_4)$.
So my question reduces to the above steps. Can you help me prove or disprove them?
Thank you!