An inheritance criterion for the admissible inference rules of $K4$

We give a criterion for inheritance of the admissible inference rules of the modal logic $K4$ by the finitely approximable modal logics extending $K4$. We give a negative solution to the question of inheritance of the admissible rules of $K4$ by tabular logics. We exhibit a series of examples of modal logics which inherit or fail to inherit the admissible inference rules of $K4$.