Can we realize Weyl group as a subgroup?

In general it is not possible to embed the Weyl group $W$ in the group $G$: already you can see this for $SL_2(mathbb C)$, where the Weyl group has order $2$: if the torus fixes the lines spanned by $e_1$ and $e_2$ respectively, you want to pick the linear map taking $e_1$ to $e_2$ and $e_2$ to $e_1$, but this has determinant $-1$. A lift of $W$ to $N(T)$ must be an element of order $4$ not $2$, say $e_1 mapsto -e_2$ and $e_2 mapsto e_1$.

In fact, Tits has shown that this is essentially the only obstruction: the Weyl group can always be lifted to a group $tilde{W}$ inside $G$ which is an extension of $W$ by an elementary abelian $2$-group of order $2^l$ where $l$ is the number of simple roots. If I recall correctly, this lift is then unique up to conjugation.