In this paper we determine all finite $2$-groups of class $2$ in which every automorphism of order $2$ leaving the Frattini subgroup elementwise fixed is inner.

In this paper we determine all finite $2$-groups of class $2$ in which every automorphism of order $2$ leaving the Frattini subgroup elementwise fixed is inner.

For any given finite abelian group, we give factorizations of the group determinant in the group algebra of any subgroups. The factorizations is an extension of Dedekind's theorem. The extension leads to a generalization of Dedekind's theorem.