  Let $K$ be a number field and $\mathcal{C}$ a full class of finite groups. We
write $K^{\mathcal{C}}/K$ for the maximal pro-$\mathcal{C}$ Galois extension of
$K$, and $G_K^{\mathcal{C}}$ for its Galois group. In this paper, we deal with
the following question: ``For $i=1,2$, let $K_i$ be a number field,
$\mathcal{C}_i$ a nontrivial full class of finite groups, and $\sigma
:G_{K_1}^{\mathcal{C}_1}\overset \sim \rightarrow G_{K_2}^{\mathcal{C}_2}$ an
isomorphism. Is $\sigma$ induced by a unique isomorphism between
$K_2^{\mathcal{C}_2}/K_2$ and $K_1^{\mathcal{C}_1}/K_1$?'' In one of the main
results, we answer this question affirmatively only assuming that the upper
Dirichlet density of the set of prime numbers concerning $\mathcal{C}_i$ is not
zero for at least one $i$. Moreover, we obtain some results which are still
valid even when $\mathcal{C}_1$, $\mathcal{C}_2$ consist of all finite
$p$-groups for a prime number $p$, that is, $G_{K_1}^{\mathcal{C}_1}$,
$G_{K_2}^{\mathcal{C}_2}$ are the maximal pro-$p$ quotients of the absolute
Galois groups.
