Component Groups of Reductive Groups

Question

Suppose $G$ is a reductive group that is not necessarily connected and $Z \subset G$ is a central subgroup. Suppose $G^0$ is the identity component of $G$. Is it true that $G/G^0Z= \pi_0(G/Z)$? I can see there is a surjection $G \twoheadrightarrow \pi_0(G/Z)$ which induces a surjection $G/G^0Z \twoheadrightarrow \pi_0(G/Z)$ but I don't see why this has to be injective.