Let $G$ be a connected split reductive group over a finite field $k$. Suppose $G$ has connected centre. Let $T$ be a maximal split torus with Weyl group $W$. Note that $W$ acts on the finite group $T(k)$; thus, it acts on characters of $T(k)$.

Let $\theta: T(k) \rightarrow \mathbb{C}^\times$ be a $W$-invariant character.

Question: Does $\theta$ extend to a character of $G(k)$?