Are there some relationship between mapping the bordism invairants of eq.1 and eq.2? $$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$ and $$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \tag{eq.2}$$ Here $K(G,2)$ is the Eilenberg–MacLane space. We can take $d=3$ here.

The semi-direct product of $q \in \mathbb{Z}_2$ here acts on the $g \in PSU(2^n)$ as the complex conjugation $*$ and transpose $T$, so $$q g q= g^{*T}$$

When $n=1$, we can simply take that eq.1's as direct product: $$\Omega_{O}^3(B(PSU(2)\times\mathbb{Z}_2))$$




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