# Relating bordism invairants in $d$ and $d+2$ dimensions

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: $$\Omega_{O}^3(B(PSU(2)\times\mathbb{Z}_2))$$