Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like $${\Omega_d^{(\mathrm{spin} \times G)/N}},$$ where $N$ is a normal subgroup shared by the $\operatorname{spin}$ and $G$ (e.g. $N=\mathbb{Z}_2$ such that $N \subset \operatorname{spin}(d)$ and $N \subset G$.) What is the difference/comparison/discrepency between Spin cobordism v.s. KO theory, do we have $$\Omega_d^{(\mathrm{spin} \times G)/N} = \text{certain } ko \text{ theory?}$$ or generally $$\Omega_d^{(\mathrm{spin} \times G)/N} \neq \text{certain } ko \text{ theory},$$ what are the differences in low dimensions (say $d \leq 7$?) or maybe other dimensions?
