I am studying about the structure of orthogonal groups and struggling to understand the relations between groups in the title:
From algebraic groups point of view, it is known that $Spin_n(q)$ is the univesal cover of the simple orthogonal group $P\Omega_n(q)$, that is, $Spin_n(q)$ is the biggest perfect group such that $Spin_n(q)/Z(Spin_n(q)) \cong P\Omega_n(q)$ (*).
Moreover, by the short exact sequence $1 \rightarrow \mathbb{Z}_2 \rightarrow Spin_n(q) \rightarrow SO_n(q) \rightarrow 1 $, we have $Spin_n(q)/\mathbb{Z}_2 \cong SO_n(q)$ (**).
This is while $SO_n(q)$ is not generally a perfect group and it has a subgroup of index 2, which is its derived subgroup $\Omega_n(q)$. Even more, the projective image of $SO_n(q)$ by its center is not generally a simple group. Therefore, using (**), we deduce that
$(Spin_n(q)/\mathbb{Z}_2) / (Z(Spin_n(q))/\mathbb{Z}_2) \cong SO_n(q)/Z(SO_n(q))=PSO_n(q)$,
which is not generally simple. This seems to be in contradiction with (*).
Now my question is that how can we explain (*) and (**) together, for example in cases when $n$ is odd.