I'm studying Pin cobordism groups of a point for some low dimensions. I found a general result by Anderson, Brown, Peterson in Theorem 5.1 of their paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. Using the Pontryagin-Thom construction, they are able to get the following results:

1) The contribution to $\Omega_{*}^{Pin}$ of terms $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{K}(\mathbb{Z}_2,n))$ is a direct summand of $\mathbb{Z}_2$ in each dimension $\geq n$. 2) The contribution to $\Omega^{Pin}_*$ of terms $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{B}O \langle 8n \rangle)$ is as follows: $\mathbb{Z}_2$ in dim$8n+i, i \equiv 0, 1(8)$; 0 in dim$8n+i, i\equiv 3, 4, 5, 7(8)$; $\mathbb{Z}_{2^{4k+3}}$ in dim$8n+8k+2, k\geq0$; and $\mathbb{Z}_{2^{4k+4}}$ in dim$8n+8k+6, k\geq0$. 3) The contribution to $\Omega^{Pin}_{*}$ of terms $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{B}O \langle 8n+2 \rangle)$ is as follows: $\mathbb{Z}_2$ in dim$8n+2+i, i\equiv 1, 2, 5, 7(8)$; $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ in dim$8n+2+i, i\equiv6(8)$; 0 in dim$8n+2+i\equiv3(8)$; $\mathbb{Z}_{2^{4k+1}}$ in dim$8n+2+8k$, $k\geq0$; and $\mathbb{Z}_{2^{4k+2}}$ in dim$8n+2+8k+4$, $k\geq0$.

However, when I tried to apply the above theorem for some low dimensional cases, I get some apparent contradictions. For example, in dim=2, if we apply the above theorem, we get $\Omega^{Pin}_2 = \mathbb{Z}_8 \oplus \mathbb{Z}_2$, which contradicts the known result of $\mathbb{Z}_8$. (Somehow by considering contributions only from $\pi_{*}(\mathbb{RP}^{\infty} \wedge \textbf{B}O \langle 8n \rangle)$ and ignoring all other contributions, one gets the correct results up to dimension 7.) Also, for dim=22, when applying this theorem, one gets $\Omega^{Pin}_{22} = \mathbb{Z}_{2^{12}} \oplus \mathbb{Z}_{2^8} \oplus \mathbb{Z}_{2^4} \oplus \mathbb{Z}_{2^{10}} \oplus \mathbb{Z}_{2^6} \oplus \mathbb{Z}_{2^2}$, instead of $\Omega^{Pin}_{22} = \mathbb{Z}_{2^{12}} \oplus \mathbb{Z}_{2^8} \oplus \mathbb{Z}_{2^6} \oplus \mathbb{Z}_{2^4} \oplus \mathbb{Z}_{2^4} \oplus \mathbb{Z}_{2^2} \oplus \mathbb{Z}_{2^2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$, which is given as an example on page 467 of their paper.

Can anyone make sense of the above theorem?