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 $\theta\in\rm Irr(N)$ is $G$-invariant. If $G/N=Q_8$
show that $θ$ extends to $G$.
( Hint : We may assume that $N\subseteq Z(G)$ and that $\theta$ is faithful. Write
$Z/N=Z(G/N)$,$G/N=<Nx, Ny>$ 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 Issacs, 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 $\theta\in \rm Irr(N)$ 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 $\beta=1$.

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 $(G,N,θ)$ is isomorphic to some
$(G^∗,N^∗,θ^∗)$, where $N^∗⊆Z(G^∗)$, and $θ^∗$ is linear and faithful.