The paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. only gives the homotopy groups of $RP^{\infty }\wedge X$ where $X$ is either $\Sigma ^nHZ/2$, $bo\langle 8n\rangle$ and $bo\langle 8n+2\rangle$, noting that they are summands of the homotopy groups of the spectrum $MPin$. However, it doesn't say how many copies of them actually show up. To find it out, you have to refer to the paper "Structure of Spin Cobordism Ring" by the same authors, Theorem 2.2 (and the definition of $J$'s on the top of the same page).

Thus $RP^{\infty }\wedge bo\langle 8n\rangle$ appears as many times as the number of the sequences $(j_1,\cdot j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n$

and $RP^{\infty }\wedge bo\langle 8n+2\rangle$ appears as many times as the number of the sequences $(j_1,\cdot j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n+1$

For the latter here is no such sequence when $n=0$, this is why there is no contribution of $bo\langle -2\rangle$.

As to the case of dimension 22 there will be one copy of $bo\langle 0\rangle$, one copy of $bo\langle 8\rangle$ and two copies of $bo\langle 16\rangle$ (one for the sequence $ (2,2)$ and another for $(4)$) as well as a copy of $bo\langle 10\rangle$ and two copies of $bo\langle 18\rangle$. Thus there is no contradiction.

The paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. only gives the homotopy groups of $RP^{\infty }\wedge X$ where $X$ is either $\Sigma ^nHZ/2$, $bo\langle 8n\rangle$ and $bo\langle 8n+2\rangle$, noting that they are summands of the homotopy groups of the spectrum $MPin$. However, it doesn't say how many copies of them actually show up. To find it out, you have to refer to the paper "Structure of Spin Cobordism Ring" by the same authors, Theorem 2.2 (and the definition of $J$'s on the top of the same page).

Thus $RP^{\infty }\wedge bo\langle 8n\rangle$ appears as many times as the number of the sequences $(j_1,\cdots ,j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n$

and $RP^{\infty }\wedge bo\langle 8n+2\rangle$ appears as many times as the number of the sequences $(j_1,\cdots ,j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdots ,+j_k=2n+1$

For the latter here is no such sequence when $n=0$, this is why there is no contribution of $bo\langle -2\rangle$.

As to the case of dimension 22 there will be one copy of $bo\langle 0\rangle$, one copy of $bo\langle 8\rangle$ and two copies of $bo\langle 16\rangle$ (one for the sequence $ (2,2)$ and another for $(4)$) as well as a copy of $bo\langle 10\rangle$ and two copies of $bo\langle 18\rangle$. Thus there is no contradiction.

The paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. only gives the homotopy groups of $RP^{\infty }\wedge X$ where $X$ is either $\Sigma ^nHZ/2$, $bo\langle 8n\rangle$ and $bo\langle 8n+2\rangle$, noting that they are summands of the homotopy groups of the spectrum $MPin$. However, it doesn't say how many copies of them actually show up. To find it out, you have to refer to the paper "Structure of Spin Cobordism Ring" by the same authors, Theorem 2.2 (and the definition of $J$'s on the top of the same page).

Thus $RP^{\infty }\wedge bo\langle 8n\rangle$ appears as many times as the number of the sequences $(j_1,\cdot j_k)$, $j_i\geq 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n$

and $RP^{\infty }\wedge bo\langle 8n-2\rangle$ appears as many times as the number of the sequences $(j_1,\cdot j_k)$, $j_i\geq 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n+1$

For the latter here is no such sequence when $n=0$, this is why there is no contribution of $bo\langle -2\rangle$.

As to the case of dimension 22 there will be one copy of $bo\langle 0\rangle$, one copy of $bo\langle 8\rangle$ and two copies of $bo\langle 16\rangle$ (one for the sequence $ (2,2)$ and another for $(4)$) as well as a copy of $bo\langle 10\rangle$ and two copies of $bo\langle 18\rangle$. Thus there is no contradiction.

The paper "Pin cobordism and related topics" http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42. only gives the homotopy groups of $RP^{\infty }\wedge X$ where $X$ is either $\Sigma ^nHZ/2$, $bo\langle 8n\rangle$ and $bo\langle 8n+2\rangle$, noting that they are summands of the homotopy groups of the spectrum $MPin$. However, it doesn't say how many copies of them actually show up. To find it out, you have to refer to the paper "Structure of Spin Cobordism Ring" by the same authors, Theorem 2.2 (and the definition of $J$'s on the top of the same page).

Thus $RP^{\infty }\wedge bo\langle 8n\rangle$ appears as many times as the number of the sequences $(j_1,\cdot j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n$

and $RP^{\infty }\wedge bo\langle 8n+2\rangle$ appears as many times as the number of the sequences $(j_1,\cdot j_k)$, $j_i> 1$, $k\geq 0$ ($k=0$ means that we have an empty sequence) with $j_1+\cdot +j_k=2n+1$

For the latter here is no such sequence when $n=0$, this is why there is no contribution of $bo\langle -2\rangle$.

As to the case of dimension 22 there will be one copy of $bo\langle 0\rangle$, one copy of $bo\langle 8\rangle$ and two copies of $bo\langle 16\rangle$ (one for the sequence $ (2,2)$ and another for $(4)$) as well as a copy of $bo\langle 10\rangle$ and two copies of $bo\langle 18\rangle$. Thus there is no contradiction.