It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.

If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$

Question 1: Then do we have $$\Omega_d^{spin}(BG)_p = ko_d(BG)_p?$$ for $p=2$ and free part, for $d\le 7$.

And $$ \Omega_d^{spin}(BG)_p = \Omega_d^{SO}(BG)? $$ for $p \neq 2$ and $p$ is an odd prime.

Namely, the 2-torsion and free part of $Mspin$ and $KO$ is the same. If there is an odd $p$ torsion, we need to consider localization at odd prime by $MSO$ cohomology. Is this correct?

Question 2: If this is a statement about the spectra, not just about stable homotopy groups, and thus within these spin cobordism and ko theory, do they completely coincide for any dimensions $d$, instead of just $d \leq 7$?