Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ and $\eta$ be the generic point. Then $G=Gal(\overline\eta/\eta)$ is isomorphic to the Galois group of the local field at $s$.

For any scheme $X\to S$, the nearby cycle functor is a functor $$R\Phi_X:D^b_c(X_{\eta},\mathbb{Q}_l)\longrightarrow D^b_c(X_{\bar s},\mathbb{Q}_l)$$ and carrying an action of $G$ compatible with the action on $X_{\overline s}$. Recall that we have an exact sequence $$0\to I\to G\to Gal(\overline s/s)\to 1$$ where $I$ is the inertia subgroup, and $I$ carries a surjective homomorphism $$t_l:I\longrightarrow \mathbb{Z}_l(1).$$ My question is: Is it true that the action of $G$ on $R\Phi_X$ always factor through $G\longrightarrow \mathbb{Z}_l(1)\rtimes Gal(\overline s/s)$?

Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ and $\eta$ be the generic point. Then $G=Gal(\overline\eta/\eta)$ is isomorphic to the Galois group of the local field at $s$.

For any scheme $X\to S$, the nearby cycle functor is a functor $$R\Phi_X:D^b_c(X_{\eta},\mathbb{Q}_l)\longrightarrow D^b_c(X_{\bar s},\mathbb{Q}_l)$$ and carrying an action of $G$ compatible with the action on $X_{\overline s}$. Recall that we have an exact sequence $$0\to I\to G\to Gal(\overline s/s)\to 1$$ where $I$ is the inertia subgroup, and $I$ carries a surjective homomorphism $$t_l:I\longrightarrow \mathbb{Z}_l(1).$$ My question is: When is it true that the action of $G$ on $R\Phi_X$ factor through $G\longrightarrow \mathbb{Z}_l(1)\rtimes Gal(\overline s/s)$?