= Solution
We use complex representations. By <Maschke's theorem>, restriction to any finite subgroup is a <semisimple module>, so it has a canonical <isotypic decomposition>. In particular, the $\theta$-<isotypic component> $W_\theta$ is the sum of all copies of the irreducible $N$-module with character $\theta$. The central <character idempotent>
$$
e_\theta=\frac{\theta(1)}{|N|}\sum_{n\in N}\theta(n^{-1})n
$$
projects onto this component. These projections preserve every $N$-submodule, which consequently decomposes as the <direct sum> of its intersections with the <isotypic components>.
Let $W$ afford $\xi\in\operatorname{Irr}(T\mid\theta)$. The component $W_\theta$ is nonzero. Since $T=I_G(\theta)$ fixes $\theta$, it preserves $W_\theta$. Irreducibility of $W$ as a $T$-module therefore gives $W=W_\theta$: its restriction to $N$ consists entirely of copies of $\theta$.
Form the <induced representation>
$$
V=\mathbb C[G]\otimes_{\mathbb C[T]}W=\bigoplus_{g\in\mathcal R}g\otimes W,
$$
where $\mathcal R$ represents the left cosets $G/T$. Because $N\triangleleft G$, for $n\in N$ we have
$$
n(g\otimes w)=g\otimes(g^{-1}ng)w.
$$
Thus $g\otimes W$ is $N$-isotypic of type $\theta^g$, where $\theta^g(n)=\theta(g^{-1}ng)$. These types are distinct for distinct cosets $gT$, precisely by the definition of the <inertia group of a character> $T$.
Let $Y$ be a nonzero $G$-submodule of $V$. Its <isotypic decomposition> as an $N$-module shows that it meets some $g\otimes W$ nontrivially. Acting by $g^{-1}$ gives $Y\cap(1\otimes W)\ne0$. This intersection is a $T$-submodule of the irreducible $W$, hence equals $1\otimes W$. The $G$-translates of that component span $V$, so $Y=V$. The induced representation is therefore irreducible, and its restriction contains $\theta$. Its <induced character> satisfies
$$
\boxed{\xi^G\in\operatorname{Irr}(G\mid\theta).}
$$
This proves the irreducibility assertion directly from <isotypic components>; no form of the correspondence being proved has been assumed.
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