Boundary normal subgroups in Clifford theory

For $N=1$, the trivial group has just its one-dimensional trivial irreducible: every subspace is invariant, so a nonzero irreducible space has dimension one. Every $G$-module restricts to its dimension many copies of this type. Conjugation fixes it, so $I=G$, its isotypical component is the whole module, and induction from $G$ to itself is identity. The ramification is therefore $\chi (1)$.

The trivial normal type extends to the trivial representation of $G$. Gallagher with $G\mathrm{/}1=G$ sends $\eta$ to $1\cdot \eta =\eta$, recovering all of $\mathrm{Irr}(G)$. A higher-degree irreducible thus has homogeneous but reducible restriction to $1$.

For $N=G$, characters are fixed by inner conjugation, so $I=G$. The restriction of an irreducible to the same group is itself, hence it contains $\theta$ exactly when it equals $\theta$, then with multiplicity one. Clifford correspondence is the singleton identity. The module affording $\theta$ extends to itself, and Gallagher has the trivial quotient $G\mathrm{/}G$, again giving only $\theta$. If $G=1$, these two endpoint descriptions agree and all degrees and indices are one.