Define a *Chebotarëv datum* over a number field $K$ to be a finite group $G$ together with a map $\mathfrak{p}\mapsto\gamma_{\mathfrak{p}}$ from a cofinite set of primes of $K$ into the set of conjugacy classes of $G$ such that for every conjugacy class $c\subset G$, the proportion of $\mathfrak{p}$ with $\gamma_{\mathfrak{p}}=c$ is $\operatorname{Card}c/\operatorname{Card}G$.  

Two Chebotarëv data $(G,\gamma)$, $(G',\gamma')$ over the same number field are said to be *equivalent* if there is an isomorphism $\varphi:G\to G'$ such that $\varphi(\gamma_{\mathfrak{p}})=\gamma'_{\mathfrak{p}}$ for almost all $\mathfrak{p}$.  If so, we identify the two.

Every finite galoisian extension $L$ of $K$ gives rise to a Chebotarëv datum $(\operatorname{Gal}(L|K),\gamma_{L|K})$ (*Chebotarëv's density theorem*).

Moreover, if $L_1$ and $L_2$ are two finite galoisian extensions of $K$ for which the associated Chebotarëv data $(\operatorname{Gal}(L_i|K),\gamma_{L_i|K})$ are equivalent, then $L_1=L_2$ (see Lemma 1, p.20, of [Mazur's recent article][1]).

**Question.**  Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?


  [1]: http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2011-01326-X/home.html