Consider a sufficiently nice topological space $X$ as well as topological groups $G$ and $H$. Consider the functor $F$ that associates to $X$ the set of all isomorphism classes of all principal $H$-bundles $\pi_H\colon P_H\twoheadrightarrow P_G$ on all principal $G$-bundles $\pi_G\colon P_G\twoheadrightarrow X$ on $X$. I think that this would be representable by the Brown representability theorem, i.e. there is a topological space $C_{G,H}$ such that $[X,C_{G,H}]$ is in bijection with the isomorphism classes of such bundles-over-bundles (modulo pointed-vs-unpointed spaces). But is there an explicit description/characterisation of it?

I am specifically interested in the case where $G=\operatorname U(1)^n$ is a torus (and $H$ is compact Lie); the motivation is that this would describe the so-called Kaluza–Klein reductions of gauge fields on nontrivially fibred torus compactifications.

As a special case, the subset of $F(X)$ that corresponds to $H$-bundles on the trivial $n$-torus bundle $\operatorname U(1)^n\times X$ is classified by the $n$-fold iterated free loop space $\mathcal L^n\mathrm BG$; and in the case $n=1$ this object is well studied. But I don't know how to generalise to arbitary principal torus bundles.