Kernels of Toeplitz operators and rational interpolation

The kernel of a Toeplitz operator on the Hardy class $H^2$ in the unit disk is a nearly invariant subspace of the backward shift operator, and, by D. Hitt's result, it has the form $g\cdot K_\omega$, where $\omega$ is an inner function, $K_\omega=H^2\ominus\omega H^2$, and $g$ is an isometric multiplier on $K_\omega$. We describe the functions $\omega$ and $g$ for the kernel of the Toeplitz operator with symbol $\bar\theta\Delta$, where $\theta$ is an inner function and $\Delta$ is a finite Blaschke product.