  Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is
cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$
which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of
$F$. In this work, we study the finitude of the Galois group
$G_{\{\mathfrak{q}\}}(F)$ of $F_{\{\mathfrak{q}\}}$ over $F$. We prove that
$G_{\{\mathfrak{q}\}}(F)$ is finite for the majority of $\mathfrak{q}$'s such
that the generator rank of $G_{\{\mathfrak{q}\}}(F)$ is two, provided that for
$p = 3$, $F$ is not a complex quartic field containing the primitive third
roots of unity.
