  Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation
Problem for $G$ asks whether the restriction map
$H^1(k,G)\to\prod_{v\in\Sigma}H^1(k_v,G)$ is surjective for every finite set of
places $\Sigma\subseteq\Omega_k$ disjoint from $\text{Bad}_G$, where
$\text{Bad}_G$ is the finite set of places that either divides the order of $G$
or ramifies in the minimal extension splitting $G$. In this paper we prove that
the set $\text{Bad}_G$ is "sharp". To achieve this we prove that there are
finite abelian $k$-groups $A$ where the map
$H^1(k,A)\to\prod_{v\in\Sigma_0}H^1(k_v,A)$ is not surjective in a set
$\Sigma_0\subseteq\text{Bad}_A$ with particular properties, namely $\Sigma_0$
is the set of places that do not divide the order of $A$ and ramify in the
minimal extension splitting $A$.
