Let $q$ be a power of a fixed prime number $p$, and let $\mathbb{F}_q$ be the finite field of $q$ elements. For a geometrically connected algebraic curve $X$ defined over $\mathbb{F}_q$, we will denote by $\pi_1(X)$ the arithmetric fundamental group of $X$.
Now suppose that $G$ is a profinite group which is isomorphic to $\pi_1(X)$ for some smooth geometrically connected algebraic curve $X$ defined over $\mathbb{F}_q$. My question is the following.
If $H$ is an open subgroup of $G$, then is there some smooth geometrically connected algebraic curve $Y$ defined over $\mathbb{F}_q$ such that $\pi_1(Y)\cong H$?
My motivation for asking this question is that such fact holds for global number fields. That is, if $K$ is a number field, $S$ a finite set of primes of $K$ containing all all archimedean primes of $ K $ and $G_{K,S}$ the Galois group of the maximal extension of $K$ inside a fixed algebraic closure of $K$ which is unramified outside $S$, then any open subgroup of $G_{K,S}$ also has this form. In this sense, I'm asking if the analogy of the above fact for function fields holds.