Nearly invariant subspaces and rational interpolation

Given an inner function $\theta$ in the upper half-plane, consider the subspace $H^2\ominus\theta H^2$ of the Hardy space $H^2$. For a finite collection $\Lambda$ of points on the complex plane, the subspace of functions from $K_\theta$ that vanish on $\Lambda$ can be represented in the form $g\cdot K_\omega$, where $\omega$ is an inner function and $g$ is an isometric multiplier on $K_\omega$. We obtain a description of the functions $\omega$ and $g$ in terms of $\theta$ and $\Lambda$.