  In this paper, we consider the inverse Galois problem with described inertia
behavior. For a finite group $G$, one of its subgroups $I$ and a prime integer
$p$, we ask whether or not $G$ and $I$ can be realized as the Galois group and
the inertia subgroup at $p$ of an extension of $\mathbb{Q}$. We first discuss
the result when $G$ is an abelian group. Then in the case that $G$ is of odd
order, Neukirch showed that there exists such an extension if and only if the
given inertia condition is realizable over $\mathbb{Q}_p$, from which we obtain
the answer for this case by studying the structure of extensions of
$\mathbb{Q}_p$ and applying techniques from embedding problems. As a corollary,
we give an explicit presentation of the Galois group of the maximal pro-odd
extension of $\mathbb{Q}_p$. When $G=\operatorname{GL}_2(\mathbb{F}_p)$ for an
odd prime $p$, we relate our realizability problem to modular Galois
representations and use elliptic curves to give answers for those subgroups $I$
corresponding to weight 2 modular forms. Finally, we provide an example arising
from Grunwald-Wang's counterexample for which the local-global principle of our
realizability problem fails.
