  We study the trace form $q_L$ of $G$-Galois algebras $L/K$ when $G$ is a
finite group and $K$ is a field of characteristic different from $2$. We
introduce in this paper the category of $2$-reduced groups and, when $G$ is
such a group, we use a formula of Serre to compute the second Hasse-Witt
invariant of $q_L$. By combining this computation with work of Quillen we
determine the isometry class of $q_L$ for large families of $G$-Galois algebras
over global fields. We also indicate how our results generalize to Galois
$G$-covers of schemes.
