Character extension
= Character extension
{title2=$\chi_N=\theta$}
An extension of $\theta\in\operatorname{Irr}(N)$ from $N\triangleleft G$ is a character $\chi$ of $G$ whose restriction is exactly $\theta$. It is necessarily irreducible, because a nontrivial invariant subspace for $G$ would also be one for $N$. An ordinary extension can fail to exist even when $\theta$ is invariant under $G$.