In this paper, we study a class of second-order accurate and energy-stable numerical schemes for the Cahn-Hilliard model. These schemes are constructed by combining the Crank-Nicolson approximation with three stabilization terms in time and employing the Fourier pseudo-spectral method in space. This class of schemes includes the second-order schemes presented in previous works while providing new schemes by introducing stabilization terms. To solve these schemes with strong nonlinearity efficiently, we propose a linear iteration algorithm and prove that the algorithm satisfies a contraction mapping property in the discrete $l^4$ norm. Furthermore, we establish a comprehensive theoretical analysis, including unique solvability, mass conservation, energy stability, and convergence based on a uniform-in-time $l^∞$ bound of the numerical solution for the proposed second-order scheme. Some numerical simulation results with many different sets of stabilization parameters are presented to conclude the paper.