An inverse scattering problem is considered for arbitrarily shaped cylindrical objects that have inhomogeneous impedance boundaries and are buried in arbitrarily shaped cylindrical dielectrics. Given the shapes of the impedance object and the dielectric, the inverse problem consists of reconstructing the inhomogeneous boundary impedance from a measured far field pattern for an incident time-harmonic plane wave. Extending the approach suggested by Akduman and Kress [Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape. Radio Sci. 38 (2003), pp. 1055-1064] for an impedance cylinder in an homogeneous background medium, both the direct and the inverse scattering problem are solved via boundary integral equations. For the inverse problem, representing the scattered field as a potential leads to severely ill-posed linear integral equations of the first kind for the densities. For their stable numerical solution Tikhonov regularization is employed. Knowing the scattered field, the boundary impedance function can be obtained from the boundary condition either by direct evaluation or by a least squares approach. We provide a mathematical foundation of the inverse method and illustrate its feasibility by numerical examples.