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.