How can i find the kernel of this integral transform?

Question

I'm trying to define a class of integral transforms $\mathfrak{S}:\mathcal{C}^\infty\rightarrow\mathcal{C}^\infty$ with the following property: $$\mathfrak{S}_{\psi}\{f(x)\psi(x)\}(t)=\alpha_f(t)\mathfrak{S}_\psi\{\psi(x)\}(t)$$ With $\alpha_f\in\mathcal{C}^\infty$ for any $f\in\mathcal{C}^\infty$. Such an integral transformation should obviously have the form: $$\mathfrak{S}_\psi\{\Psi_f(x)\}(t)=\int_a^bf(x)\psi(x)K_f(x,t)dx$$ For some kernel $K_f$. Also, the integral's kernel should satisfy: $$\int_a^bf(x)\psi(x)K_f(x,t)dx=\alpha_f(t)\int_a^b\psi(x)K_f(x,t)dx$$ For some $\alpha_f$. Any ideas for the kernel?