On the Tannakian group scheme of $\mathcal{D}$-modules

We prove a Lefschetz theorem for the Tannakian group scheme of $\mathcal{D}$-modules, in arbitrary characteristic. In characteristic $0$, We prove a Künneth formula for the Tannakian group scheme of regular singular integrable connections, and disprove it for the Tannakian group scheme of all integrable connections without any regularity assumption.