Background: I think, Dold manifold and Wu manifold are 5-dimensional manifolds which are cobordant to each other via 5-dimensional bordism group: $$ \Omega^{SO}_5. $$
Literally, cobordism theories are represented by Thom spectra $MG$: given a group $G$, the Thom spectrum is composed from the Thom spaces $MG_n$ of the standard vector bundles over the classifying spaces $BG_n$. So we may also say that Dold manifold and Wu manifold are 5-dimensional manifolds which are cobordant to each other via 5-dimensional bordism group $$ \Omega^{}_5(BSO), $$ where $BSO$ is the classifying space of $SO$. Here we can simply take the special orthogonal group $SO=SO(5)$ since we are in 5-dimension.
Consider the following new classifying space $BG'$ constructed from the fibrations $$ K(\mathbb{Z}_2,2)\to BG' \to BSO, $$ or equivalently, $$ K(\mathbb{Z}_2,2)\to BG' \to K(SO,1), $$ where $K(\mathbb{Z}_2,2)$ is the Eilenberg–MacLane space. The possible classes of fibrations are classified by Postinikov classes $[\omega]\in H^3(BSO,\mathbb{Z}_2)=\mathbb{Z}_2.$ So there are actually two different fibrational classifying space $BG'$.
Let us consider these two new classifying space $BG'$ (denoted as $BG'_1$ and $BG'_2$.)
Questions:
Are Dold manifold and Wu manifold (5-dimensional manifolds) which are cobordant via 5-dimensional bordism group: $$ \Omega^{}_5(BG')? $$ for either $BG'_1$ and $BG'_2$.
If the above answer is yes, does it mean that (Dold manifold) $\#$ (Wu manifold) is null-bordant via $\Omega^{}_5(BG')?$ for which $BG'_1$ and $BG'_2$?