The main result of the paper reads as follows. Let \(G\) be a finite group, \(H\) a subgroup of \(G\), and let \(V\) and \(W\) \(\mathbb{C} G\)-modules affording the characters \(\chi\) and \(\psi\). (i) If \(\chi|_H\) is irreducible, then \[ \tfrac 1{|H|}\sum_{h\in H}\chi(hx)\overline{\chi(hy)}=\tfrac{\chi(xy^{-1})}{\chi(1)} \] for all \(x,y\in G\). \[ \tfrac 1{|H|}\sum_{h\in H}\chi(hx)\overline{\psi(hy)}=0\tag{ii} \] for all \(x,y\in G\) if and only if \(V|_H\) and \(W|_H\) have no common constituents. Several examples are given and interesting corollaries are derived. Among them, a generalization of Burnside's result on the existence of zeros for nonlinear irreducible characters is deduced.