Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be the full subcategory of connections that are regular at infinity.
Both $\mathscr{C}$ and $\mathscr{C}'$ are neutural Tannakian categories, with fiber functors $\omega_x$, $\omega'_x$ ($x\in X(\mathbb{C})$) taking a vector bundle to its fiber over $x$. The affine group-scheme $\underline{\mathrm{Aut}}^\otimes \omega_x'$ is identified with the algebraic envelope of $\pi_1(X(\mathbb{C}),x)$. Connections that aren't regular at infinity can be pathological, so I'm curious:
Is there a concrete description of $\underline{\mathrm{Aut}}^\otimes \omega_x$?
