Minimal supplements of maximal tori in their normalizers for the groups $F_4(q)$

Let $G$ be a finite group of Lie type $F_4$ and $W$ the Weyl group of $G$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement of $T$ in its algebraic normalizer $N(G,T)$. In particular, we find all the maximal tori that have a complement in $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. We find the minimal orders of the lifts of the elements $w$ in $N(G,T)$.