  Let $G$ be a finite group and let $k$ be an algebraically closed field of
characteristic $2$ and let $M$ be an indecomposable $kG$-module which affords a
non-degenerate $G$-invariant symmetric bilinear form. We introduce the
symmetric vertices of $M$. Each of these is a $2$-subgroup of $G$ which
contains a Green vertex of $M$ with index at most $2$. If $M$ is irreducible
then its symmetric vertices are determined up to $G$-conjugacy.
  If $B$ is the real $2$-block of $G$ containing $M$, we show that each
symmetric vertex of $M$ is contained in an extended defect group of $B$.
Moreover, we characterise the extended defect groups in terms of symmetric
vertices.
  In order to prove these results, we develop the theory of involutary
$G$-algebras. This allows us to translate questions about symmetric
$kG$-modules into questions about projective modules of quadratic type.
