While reading Fulton and Harris's book 'Representation Theory', I came across Proposition 5.1, which basically states the following:
Let $V$ be an irreducible representation of $G$ and let $W$ be the representation when restricted to $H$, a normal subgroup of $G$. Let |G / H| = 2. Then, either $W$ is still irreducible or $W$ splits up into the direct sum of two irreducible representations of the same dimension that are not isomorphic.
I am confused on the second case. I believe that this will happen when $V$ is an irreducible representation for $G$, but not for $H$. How do we know $W$ will split up into two irreducible representations, that they are of the same dimension and are not isomorphic?