Squares of characters and finite groups

Abstract

Let $G$ be a group of odd order and $χ$ be a complex irreducible character. Then there exists a unique character $χ^{(2)}\in\Irr(G)$ such that $[χ^2,χ^{(2)}]$ is odd. Also, there exists a unique character $ψ\in \Irr(G)$ such that $[ψ^2, χ]$ is odd.