The pro-$\mathcal{C}$ anabelian geometry of number fields

Let $K$ be a number field and $\mathcal{C}$ a full class of finite groups. We write $K^{\mathcal{C}}\mathrm{/}K$ 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 $\sigma :G_{K_1}^{{\mathcal{C}}_1}{G}_{K_2}^{{\mathcal{C}}_2}$ an isomorphism. Is $\sigma$ induced by a unique isomorphism between $K_2^{{\mathcal{C}}_2}\mathrm{/}{K}_2$ and $K_1^{{\mathcal{C}}_1}\mathrm{/}{K}_1$?‘’ 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.