The analog of Huppert's conjecture on character codegrees

Let G be a finite group and χ be a character of G, we denote the codegree of χ to be codeg(χ)=|G:ker⁡χ|χ(1). We define σ(codeg(G))=max⁡{|π(codeg(χ))||χ∈Irr(G)} where π(codeg(χ)) is the set of prime divisors of codeg(χ) and π(G) to be the set of prime divisors of |G|. In this note, we show that there exists a fixed constant k such that |π(G)|≤kσ(codeg(G)). This answers an open question in [12].