 Induction from an abelian normal subgroup with a free character orbit

Let $N◃G$ be abelian and let a linear character $\theta$ of $N$ have stabilizer exactly $N$ under conjugation by $G$. Then ${\mathrm{Ind}}_N^G\theta$ is irreducible of degree $[G:N]$. Its value is zero outside $N$, while on $n\in N$ it is the sum of the characters in the $G$-orbit of $\theta$.