The $2$-character table does not determine a group

Frobenius had defined the group determinant of a group $G$ which is a polynomial in $n = \vert G\vert$ variables. Formanek and Sibley have shown that the group determinant determines the group. Hoehnke and Johnson show that the $3$-characters (a part of the group determinant) determine the group. In this paper it is shown that the $2$-characters do not determine the group. If we start with a group $G$ of a certain type then a group $H$ with the same $2$-character table must form a Brauer pair with $G$. A complete description of such an $H$ is available in Comm. Algebra 9 (1981), 627-640.