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

Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma:S\rightarrow S$ 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.