I read from this answer saying that Pin(p,q) and Pin(q,p) are not isomorphic. https://physics.stackexchange.com/a/634272/42982
The $\text{Pin}(p,q)$ and $\text{Pin}(q,p)$ are said to be different central extensions of $O(p,q) \simeq O(q,p)$.
Moreover, there seems to be also $\text{Pin}^+$ and $\text{Pin}^-$ versions of $\text{Pin}(p,q)$ and $\text{Pin}(q,p)$. So are there four of more Pin groups out of the central extensions of $O(p,q)\simeq O(q,p)$?
Can we write down the full extensions of exact sequences for these?