  Let $P$ be a principal indecomposable module of a finite group $G$ in
characteristic $2$ and let $\varphi$ be the Brauer character of the
corresponding simple $G$-module. We show that $P$ affords a non-degenerate
$G$-invariant quadratic form if and only if there are involutions $s,t\in G$
such that $st$ has odd order and $\varphi(st)/2$ is not an algebraic integer.
  We then show that the number of isomorphism classes of quadratic principal
indecomposable $G$-modules is equal to the number of strongly real conjugacy
classes of odd order elements of $G$.
