Coprime-degree determinant extension theorem

Suppose $N◃G$, $G/N$ is solvable, $\theta \in \mathrm{Irr}(N)$ is invariant under $G$, and $\theta (1)$ is coprime to $[G:N]$. If a linear character $\mu$ of $G$ restricts to the determinant character $\det \theta$, then there is a unique character extension $\chi \in \mathrm{Irr}(G)$ with determinant $\mu$. The coprime degree makes determinant normalization remove the ambiguity from twisting an extension by linear characters of the quotient group.