I am trying to understand proof of Prop. 5.1(page 64) from Fulton and Harris representation theory. I am unable to prove one statement in the proof. Which is
Let $V$ be an irreducible representation of a finite group $G$. Let $W$ be restriction of $V$ to a normal subgroup $H$ with index $2$. Now $W = W' \oplus W''$, where $W', W''$ are irreducible and conjugate but not isomorphic.
I am unable to understand why they are not self conjugate? The proof says,
Since $W$ is self conjugate and if $W', W''$ were self conjugate $V$ wouldn't be irreducible.
I am trying to assume $W',W''$ are self conjugate and produce a decomposition of $V$, but I couldn't. Please help me in filling details.
UPDATE: My idea is We know that $W', W''$ are different irreducible representations(if not, $<W,W>=4$, a contradiction as it's value is $2$) and as $W', W''$ are self conjugate by an element of $G\backslash H$. For any $g \in G, g.W' \subset W'$, if not the map $g$ gives an isomorphism between $W'$ and $W''$ but we started with two different representations.