Normal subgroups of fundamental groups of affine curves in positive characteristic II

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C\mathrm{/}k$ be a smooth connected affine curve. Denote by ${\pi }_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgroups $N$ of ${\pi }_1(C)$. In “Normal subgroups of fundamental groups of affine curves in positive characteristic” we proved the same result under the additional hypothesis that $k$ had countable cardinality.