I'm having some trouble in beginner's representation theory and am pretty lost about this problem:
Let ($\rho$, $V$) be a representation of $G$, so $\rho$: $G$ $\to$ $GL(V)$ is a group homomorphism. Let $H$$\subset$$G$ be a normal subgroup of index 2. Suppose $V$ is $G$-irreducible, but not $H$-irreducible. Prove tr($\rho$($g$))=0 for $g$$\notin$$H$.
I thought about starting by saying $G/H$ is isomorphic to $C_2$, and so you could say that some character
$\lambda$= \begin{cases} 1, & \text{if $g \in H$} \\ -1, & \text{if $g \notin H$} \end{cases}
But I don't think this is very helpful, and I feel like this is not the way to go about the proof. Any help would be appreciated!