ag.algebraic geometry – Maximal closed subscheme stable under the action of a finite connected group scheme

Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Ysubseteq X$ a closed irreducible subvariety. Let $G$ be a connected finite $k$-group scheme acting on $X$.

Does there exist a maximal closed subscheme $T$ of $Y$ stable under the action of $G$?

If $G$ is ├ętale, then I think one can define $T:=bigcap_{gin G}g(Y)$. In case $T$ exists, is there a similar description for it?