Admissibility and unification in the modal logics related to S4.2

We study unification and admissibility for an infinite class of modal logics. Conditions superimposed to these logics are to be decidable, Kripke complete, and generated by the classes of rooted frames possessing the greatest clusters of states (in particular, these logics extend modal logic S4.2). Given such logic $L$ and each formula $\alpha$ unifiable in $L$, we construct a unifier $\sigma$ for $\alpha$ in $L$, where $\sigma$ verifies admissibility in $L$ of arbitrary inference rules $\alpha/\beta$ with a switched-modality conclusions $\beta$ (i.e., $\sigma$ solves the admissibility problem for such rules).