On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes

In

The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513

Woodward proposed a classification of $$\mathrm{PU}(n)={\mathrm{U}(n)}/{Z(\mathrm{U}(n))}=\mathrm{SU}(n)/\mathbb{Z}_n$$ principle bundles over a 4-complex $$X$$ in terms of two characteristic classes $$q \in H^4(X,\mathbb{Z})$$ and $$t \in H^2(X,\mathbb{Z}_n)$$. These classes are related as $$\rho_{2n}q=(n+1)\mathfrak{P}(t)$$ for $$n$$ even (a similar expression holds for $$n$$ odd). Here $$\rho_{r}$$ is the $$\mod r$$-reduction of a integral class and $$\mathfrak{P}:H^2(X,\mathbb{Z}_{2s})\rightarrow H^2(X,\mathbb{Z}_{4s})$$ is the Pontraygin square. The class $$t$$ can be interpreted as the obstruction to define an $$\mathrm{SU}(n)$$ bundle and $$q$$ as an instanton/Chern number.

I have two questions:

1. For $$n=4$$, we have $$\mathrm{PSU}(4)=\mathrm{PSO}(6)$$. What are the conditions on $$t$$ and $$q$$ to be able to define an $$\mathrm{SO}(6)$$ bundle? If such a bundle can be defined, what are the relations between $$q$$, $$t$$ of the $$\mathrm{PSU}(n)$$ bundle and the characteristic classes of the $$\mathrm{SO}(6)$$ bundle (i.e., the Stiefel-Whitney classes $$w_{2,4}$$ and the Pontryagin class $$p_1$$). I believe there should be relation as $$2p_1=q$$.

2. Let $$P(n,mn)=\mathrm{SU}(mn)/\mathbb{Z}_n$$, where $$\mathbb{Z}_n$$ is a normal subgroup of the center of $$\mathrm{SU}(mn)$$. Is there a similar result for $$P(n,mn)$$ bundles? (i.e., classification, relation between the characteristic classes, specially with the instanton number). My intuition tells me that there should still be classes $$\tilde{q}\in H^4(X,\mathbb{Z})$$ and $$\tilde{t}\in H^2(X,\mathbb{Z}_n)$$ together with some new classes in some $$H^4(X,\mathbb{Z}_{l})$$ because of the $$n=4$$, $$m=2$$ case $$\mathrm{SU}(4)/\mathbb{Z}_2=\mathrm{Spin}(6)/\mathbb{Z}_2=\mathrm{SO}(6)$$.

I know there is some work on the classifying space of $$\mathrm{P}(n,mn)$$ by Xing Gu, e.g. On Topological Brauer Classes over 8-Complexes with Periods Divisible by 4 (arXiv:1803.05100), and The Topological Period-Index Problem over 8-Complexes (arXiv:1709.00787), but I don't know how to proceed from there.

Fix a finite 4-dim CW complex $$X$$.
First we study the homotopy classes of maps from $$X$$ to $$K(\mathbb{Z}/n,2)$$, the $$3$$rd stage of the Postnikov tower of $$\mathbf{B}P(n,mn)$$, the classifying space of $$P(n,mn)$$, which involves the k-invariant $$\kappa_3$$. The class $$\tilde{t}$$ is to be chosen such that its composite with $$\kappa_3$$ vanishes.
Then there are (homotopy classes of) maps from X to $$\mathbf{B}P(n,mn)[5]$$ (the $$5$$th stage of the Postnikov tower) lifting $$\tilde{t}$$. Roughly speaking, to choose one of them amounts to a choice of $$\tilde{q}$$.