Let $\mathbf{G}$ be a linear algebraic group over an algebraic closed field of characteristic $p$ and $F$ a proper frobenius map on it with fixed point group $\mathbf{G}^f=G$ such that $\overline{G}=G/Z(G)$ be a finite simple group of Lie type. We know that a maximal torus of $\overline{G}$ is defined to be $\overline{T}$ where $T=\mathbf{T}^F$ and $\mathbf{T}$ is a $F$-stable torus of $\mathbf{G}$.
Is it true that a maximal abelian subgroup $\overline{H}\subseteq\overline{G}$ where $gcd(|\overline{H}|,p)=1$ is a maximal torus of $\overline{G}$?
I may be saying something silly, but isn't for instance the subgroup $\mathbb{F}_p\simeq \begin{bmatrix}1 &t\\ 0 & 1\end{bmatrix}\hookrightarrow \textrm{SL}(2,\mathbb{F}_p)$ (for some $p>3$) a unipotent maximal abelian subgroup of that group that is not the reduction of a torus because it is not reductive? Please correct me if this is not maximal abelian.