Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups

Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma :S\to S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation $${\displaystyle {\int }_{S}f(xyt)d\mu (t)+{\displaystyle {\int }_{S}f(\sigma (y)xt)d\mu (t)=2f(x)g(y),x,y\in S,}}$$ and the Van Vleck-Wilson functional equation $${\displaystyle {\int }_{S}f(xyt)d\mu (t)-{\displaystyle {\int }_{S}f(\sigma (y)xt)d\mu (t)=2f(x)g(y),x,y\in S,}}$$ where $\mu$ is a measure that is a linear combination of Dirac measures $({\delta }_{z_i}{)}_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$. Interesting consequences of these results are presented.