# Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$

Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over $\overline{\mathbb{F}}_l$ with finite image $I$. Suppose that $V$ is primitive, i.e., is not induced from a proper subgroup of $I$. Under these assumptions, could it happen that $I$ has an element of order $l$?

In case it is of relevance: when restricted to the wild inertia such a $V$ remains irreducible and hence in particular is of $p$-power dimension.

Say $p=2$, $l=3$, and take an elliptic curve $E/{\mathbb Q}_2$ with largest possible inertia image, $I=\text{SL}(2,{\mathbb F}_3)$. Then the $3$-adic representation $V_3(E)$ is irreducible, $I$ acts faithfully and so has an element of order 3 on it, but it is not an induced representation as it is 2-dimensional and $I$ has no index 2 subgroups.