Recently, I am studying the book Navarro - 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 can prove this hint, but I don't know how to use it.
I also find a result in Character Theory of Finite Groups written by I M Isaacs, it gives a necessary and sufficient condition for an invariant irreducible character of a normal subgroup to be extendible.
(11.7) THEOREM Let $N\unlhd G$ and let $\theta\in \operatorname{Irr}(N)$ be invariant in $G$. Let $R$ be a representation affording $\theta$ and let $X$ be a projective representation of $G$ satisfying conditions (a),(b), and (c) of Theorem 11.2. Let $\alpha$ be the factor set of $X$. Define $\beta\in Z(G/N,\mathbb{C}^{\times})$ by $\beta(gN,hN)=\alpha(g,h)$. Then $\beta$ is well-defined and its image $\bar\beta\in H(G/N,\mathbb{C}^{\times})$ depends only on $\theta$. Also, $\theta$ is extendible to $G$ iff $\bar\beta=1$.
I wonder how to solve this problem by the hint. Maybe corollary 5.9 in Character Theory and the McKay Conjecture 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.
