Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ defined as the restriction of scalars (a la Weil) of $G$ from $M$ down to $F$, i.e. $H=\mathrm{Res}_{M/F} G$. If $T$ is a subgroup of $H$, under what circumstances/hypotheses is it true that there exists a subgroup $\tilde{T}\leq G$ such that $T=\mathrm{Res}_{M/F}\tilde{T}$? Any reference would be greatly appreciated.
Edit: Since I did not get any response, I have cross-posted my question on MathOverflow (see this link).