Recently, I am studying the book Character Theory and the McKay Conjecture. I am trying to solve the following exercise:

(Exercise 5.9)

Let $G$ be a finite group and $N\unlhd G$, suppose that $\theta\in\operatorname{Irr}(N)$ is $G$-invariant. If $G/N=Q_8$ show that $\theta$ extends to $G$. ( Hint : We may assume that $N\subseteq Z(G)$ and that $\theta$ is faithful. Write $Z/N=Z(G/N)$,$G/N=\langle Nx, Ny\rangle$ with $Nx^2=Ny^2$ of order $2$. Show that $Z\subseteq Z(G)$ and that $G'\cap N=1$, by proving that $[x,y]^2=1$.)

I know how to prove this hint, but I can't use it to solve this question.

Maybe corollary 5.9 is useful：

Corollary 5.9 Every character triple $(G,N,\theta)$ is isomorphic to some $(G^*,N^*,\theta^*)$, where $N^*\subseteq Z(G^*)$, and $\theta^*$ is linear and faithful.