For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface $X_g$ of genus $g$ that restrict to a given $U(n)$-bundle $P\rightarrow X$ over the basepoint. For the case $n=2$, $m=3$ we have a $U(2)$-bundle over the 5-complex $S^3\times X_g$. Such a bundle is classified by a (homotopy class of) map $f:S^3\times X_g\rightarrow BU(2)$, which in tern lifts up to the 5$^{th}$ Postnikov section of the classifying space $BU(2)^{(5)}=B_2^{(5)}$.

I identify the 4$^{th}$ Postnikov section as

\begin{equation} B^{(4)}_2=K(\mathbb{Z},2)\times K(\mathbb{Z},4) \end{equation}

and the $5^{th}$ k-invariant $k^5:B^{(4)}_2\rightarrow K(\mathbb{Z}_2,6)$ as the map representing the cohomology class

\begin{equation} x_2\times x_4 + 1\times Sq^2x_4\in H^6(K(\mathbb{Z},2)\times K(\mathbb{Z},4);\mathbb{Z}_2) \end{equation}

This map induces the principal fibration

\begin{equation} K(\mathbb{Z}_2,5)\xrightarrow{j^{(5)}}B^{(5)}_2\xrightarrow{p^{(5)}}K(\mathbb{Z},2)\times K(\mathbb{Z},4) \end{equation}

which, for the purposes of the orignal calculation, leaves us with the exact sequence

\begin{equation} 0\rightarrow H^5(S^3\times X_g;\mathbb{Z}_2)\xrightarrow{j^{(5)}_*}[S^3\times X_g,B^{(5)}_2]\xrightarrow{p^{(5)}_*} H^4(S^3\times X_g;\mathbb{Z})\oplus H^2(S^3\times X_g;\mathbb{Z})\rightarrow 0. \end{equation}

On the right, the sequence maps to $H^6(S^3\times X_g;\mathbb{Z}_2)=0$, so $p^{(5)}_*$ is surjective, whilst stability shows the map $Sq^2:H^3(S^3\times X_g;\mathbb{Z})\rightarrow H^5(S^3\times X_g;\mathbb{Z}_2)$, which is zero as this squaring operation is trivial, making $j^{(5)}_*$ injective. The sequence splits (although a full proof of this is involved) and the end result is

\begin{align} [S^3\times X_g,BU(2)]&=H^2(S^3\times X_g;\mathbb{Z})\oplus H^4(S^3\times X_g;\mathbb{Z})\oplus H^5(S^3\times X_g;\mathbb{Z}_2)\\ &=H^1(X_g)\oplus H^2(X_g)\oplus H^2(X_g;\mathbb{Z}_2)\notag\\ &=\mathbb{Z}^{2g+1}\oplus \mathbb{Z}_2.\notag\oplus \mathbb{Z}_2 \end{align}

The class in $H^2(X_g)$ is the first Chern class of the bundle $P$. The class in $H^1(X_g)$ represents a homotopy class in $\pi_3(B{\mathcal{G}}(P))$.

So the problem: How does one identify the class in $H^5(S^3\times X_g;\mathbb{Z}_2)\cong H^2(X_g;\mathbb{Z}_2)$ contributing to this decomposition? Given a map $f:S^3\times X_g\rightarrow BU(2)$ classifying the bundle $E$, will it have a nonzero class?

It is simple to generate a bundle with such a by class using the operation of $S^5$ on $S^3\times X_g$ given by the map pinching to the top cell. One can then tensor this with the pullback by the projection of a $U(1)$-bundle representing $P\rightarrow X_g$ to generate a bundle that restricts to $P$ over the basepoint.

I have tried constructing a secondary cohomology operation that detects the class, but so far have been unsuccessful.