We show that the effective axial-vector current couplings ($g_A^\mathrm{eff}$) for the neutrinoless double-$\beta$ ($0\nu\beta\beta$) and the two-neutrino double-$\beta$ ($2\nu\beta\beta$) decays are close to each other in the decay instance of $^{136}$Xe$\rightarrow$$^{136}$Ba. This is a remarkable finding because $g_A^\mathrm{eff}$ for the $0\nu\beta\beta$ decay was unknown at all. This is shown by our calculation with the transition operator perturbed by the residual nucleon-nucleon interaction. The nuclear wave functions are supplied by the quasiparticle random-phase approximation with phenomenological interactions.
The sum of the correction terms of the nuclear matrix element due to the perturbation is half the leading term in absolute value and negative. The influence of the perturbation on the half-life for the $2\nu\beta\beta$ decay is also shown to be significantly large. The more perturbation, the longer the half-life.