Let G be a group and C\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {C}$$\end{document} the field of complex numbers. Suppose σ:G→G\documentclass[12pt]{minimal}
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\begin{document}$$\sigma : G \rightarrow G$$\end{document} is an involutive endomorphism, that is, σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is an endomorphism of G and it satisfies the condition σ(σ(x))=x\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (\sigma (x)) = x$$\end{document} for all x in G. In this paper, we find the solutions f,g,h,k:G→C\documentclass[12pt]{minimal}
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\begin{document}$$f, g, h, k : G\rightarrow \mathbb {C}$$\end{document} of the equation f(xy)+g(σ(y)x)=h(x)+k(y)\documentclass[12pt]{minimal}
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\begin{document}$$f(xy) + g(\sigma (y) x) = h(x) + k(y)$$\end{document}for allx,y∈G\documentclass[12pt]{minimal}
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\begin{document}$$\text {for all } x, y \in G$$\end{document} assuming f and g to be central functions. This equation is a variant of a generalized quadratic functional equation on groups with an involutive endomorphism. As an application, using the solutions of this equation, we find the solutions f,g,h,k:G×G→C\documentclass[12pt]{minimal}
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\begin{document}$$f, g, h , k : G \times G \rightarrow \mathbb {C}$$\end{document} of the equation f(pr,qs)+g(sp,rq)=h(p,q)+k(r,s)\documentclass[12pt]{minimal}
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\begin{document}$$f(pr, qs)+g(sp,rq) = h(p,q) + k(r,s)$$\end{document} for all p,q,r,s∈G\documentclass[12pt]{minimal}
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\begin{document}$$p, q, r, s \in G$$\end{document} assuming f and g to be central functions.
