Extension in semidirect case

Question

Suppose $G=NQ$, where $N$ is normal in $G$ and $(|N|,|Q|)=1.$ In other words $G$ is a semidirect product of $N$ and $Q.$

Can we say every irreducible character of $N$ is extendable to $G?$

Suppose $\theta$ is an irreducible character of $N.$ I think if $\theta$ is $G$-invariant it's true. I can show every nonlinear irreducible character of $N$ is extendable by Clifford’s correspondence.

Answer 1

It is true if $\theta$ is $G$-invariant and $|N|$ is even (using gcd$(\theta(1),|G:N|)=1$). But conversely, if $\theta$ is extendible, it must be $G$-invariant (again, use Clifford’s Theorem).