We show that the exact beta-function (g) in the continuous two-dimensional g⁴ model possesses the Kramers-Wannier duality symmetry. The duality symmetry transformation = d(g) such that (d(g)) = d´(g)(g) is constructed and the approximate values of g^* computed from the duality equation d(g^*) = g^* are shown to agree with the available numerical results. The calculation of the beta-function (g) for the 2D scalar g⁴ field theory based on the strong-coupling expansion is developed and the expansion of (g) in powers of g^-1 is obtained up to order g^-8. The numerical values calculated for the renormalized coupling constant g₊^* are in reasonably good agreement with the best modern estimates recently obtained from the high-temperature series expansion and with those known from the perturbative four-loop renormalization-group calculations. The application of Cardy's theorem for calculating the renormalized isothermal coupling constant g_c of the 2D Ising model and the related universal critical amplitudes is also discussed.