Integral of an extension of the sine addition formula

In this paper, we determine the continuous solutions of the integral functional equation of Stetkær's extension of the sine addition law $\int_{G}f(xyt)d\mu(t)=f(x)\chi_1(y)+\chi_2(x)f(y)$, $x,y\in G$, where $f\colon G\rightarrow \mathbb{C}$, $G$ is a locally compact Hausdorff group, $\mu$ is a regular, compactly supported, complex-valued Borel measure on $G$ and $\chi_1$, $\chi_2$ are fixed characters on $G$.

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