Let $G$ be a split reductive group and $T$ its maximal torus. Is it always possible to embed $G$ into some split reductive group $G'$ such that $G,G'$ have the same unipotent $U$, same Weyl group $W$, and the unramified cocharacter group $X_*(G')$ is isomorphic to $\mathbb{Z}^n$ as a free module?
The examples I have in mind are $SL_n$ that can be embedded in $GL_n$ or $Sp_{2n}$ in $GSp_{2n}$. Is there something like this for an arbitary split reductive group or a name for this sort of relation between two such groups?