  We prove Clifford theoretic results on the representations of finite groups
which only hold in characteristic $2$.
  Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let
$\varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We
prove that there is a unique self-dual irreducible Brauer character $\theta$ of
$G$ such that $\varphi$ occurs with odd multiplicity in the restriction of
$\theta$ to $N$. Moreover this multiplicity is $1$.
  Conversely if $\theta$ is an irreducible $2$-Brauer character of $G$ which is
self-dual but not of quadratic type, the restriction of $\theta$ to $N$ is a
sum of distinct self-dual irreducible Brauer character of $N$, none of which
have quadratic type.
  Let $b$ be a real $2$-block of $N$. We show that there is a unique real
$2$-block of $G$ covering $b$ which is weakly regular.
