I've been trying to understand some of the idiosyncrasies associated to algebraic groups over non-algebraically closed fields $K$ of characteristic $p > 0$.

Let $G$ is a connected almost absolutely simple adjoint algebraic group over $K$ and $\widetilde{G}$ is the simply connected cover of $G$ also defined over $K$ with central isogeny $\pi : \widetilde{G} \rightarrow G$. Let $\widetilde{T} \subseteq \widetilde{G}$ be a maximal $K$-torus and let $T= \pi(\widetilde{T}) \subseteq G$.

Is it possible for the $\mathbb{Z}$-rank of the $K$-defined character groups $X_K(\widetilde{T})$ and $X_K(T)$ to be different? In other words, is the $K$-split part of $\widetilde{T}$ always the same rank as the $K$-split part of $T$?