Some generalization of Cauchy's and the quadratic functional equations (Q662368)

Suppose that \((S,+)\) is an abelian group and \(\Lambda\) is a finite subgroup of the automorphism group of \(S\), \(L=\text{card} \Lambda\). Let \((H,+)\) be an abelian group uniquely divisible by \((L+1)!\). The author finds the solutions \(f, , h:S \to H\) of the equation \[ \sum_{\lambda \in \Lambda}f(x+\lambda y)=Lg(x)+h(y)\,\,(x,y\in S). \]