Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$. Is it true that there exists a stratification $\tau$ of $X$ which is $G$-equivariant and such that $F$ is $\tau$-constructible? For example, one could inspect the orbit stratification.
I am trying to use the characterisation of invariant perverse sheaves as those perverse sheaves such that $act^* F\simeq pr_2^*F$ where $act:G\times_k X\to X$ is the action and $pr_2:G\times X\to X$ is the second projection. But I cannot find the solution.
Thank you in advance.