Restricted interpolation over modal logic $\mathrm S4$

The problem of restricted interpolation and definability in normal extensions of modal logic $\mathrm S4$ is investigated.We specify necessary conditions for the restricted interpolation property IPR in the systems under consideration, and prove that there exist only finitely many logics possessing IPR or the projective Beth property PB2. These logics are all residually finite and recognizable over $\mathrm S4$. As a consequence, the restricted interpolation problem and the projective Beth property are decidable over $\mathrm S4$.