Abelian intersection of irreducible inducing subgroups ($K=\bigcap \{H:{\mathrm{Ind}}_H^G\phi \text{ is irreducible for some }\phi \}$)

In a finite monomial group, the displayed intersection is a normal subgroup. Every irreducible character is induced from a linear character of a subgroup containing $K$. On restriction to $K$, all conjugate inducing lines remain invariant, so $K^{\prime }$ lies in every irreducible character kernel. The intersection of those kernels is trivial, proving that $K$ is Abelian.