By Hilbert's Theorem 90, every torsor for a split torus over a field is trivial. Thus, as pranavk has commented, this should give surjectivity.

$\textbf{Edit.}$ The first answer I wrote (now changed) applied to the full center $Z$. I did not realize that the OP is asking about the quotient by $Z_e$, the connected component of the identity. I have corrected my answer.