The paper presents the KINX code for computing linear ideal MHD growth rates and eigenvectors of axisymmetric plasmas surrounded by a vacuum layer and a conducting wall. Plasma equilibrium magnetic surfaces are assumed to be nested either in the whole plasma domain (separatrix at the plasma boundary is possible) or in the domains separated by an internal separatrix (doublet and divertor configurations). The computational domain is decomposed into subdomains with nested flux surfaces. In each subdomain finite hybrid elements are used on an equilibrium grid adapted to magnetic surfaces. Numerical destabilization is eliminated; this results in better convergence properties and makes possible efficient stability index calculation (delta W-code). An inverse vector iteration method and a vectorizable matrix solver are applied to the matrix eigenvalue problem. The stability studies of external kink modes for doublet and single null configurations are given as application examples of the KINX code. Another version of the code, KINX-W, computing resistive wall n = 0 mode growth rates, is also presented for single null, doublet and divertor plasma configurations.