In this paper, we derive a numerical solution of an inverse obstacle scattering problem with conductive boundary condition. The aim of the direct problem is the computation of the scattered field for a given arbitrarily shaped cylinder with conductive boundary condition on its surface. The inverse problem considered here is the reconstruction of the conductivity function of the scatterer from meausurements of the far field. A potential approach is used to obtain boundary layer integral equations both for the solution of the direct and the inverse problem. The numerical solutions of the integral equations which contain logarithmically singular kernels are evaluated by a Nystrom method and Tikhonov regularization is used to solve the first kind of integral equations occuring in the solution of the inverse problem. Finally, numerical simulations are carried out to test the applicability and the effectiveness of the method.