Abstract: By using the properties between the map f in metric G-space and induced map \hatf in orbital space, the dynamical relationship between G-Lipschitz shadowing property, G-equicontinuity, G-non-wandering point of the map f and Lipschitz shadowing property, equicontinuity, non-wandering point of the induced map \hatf are studied. The following conclusions are obtained: (1)The map f has G-Lipschitz shadowing property if and only if the induced map \hatf has Lipschitz shadowing property. (2)The map f is G-equicontinuous if and only if the induced map \hatf is equicontinuous. (3)The G-non-wandering point set Ω_G(f) of the map f is dense in X if and only if the non-wandering point set Ω(\hatf) of the induced map \hatf is dense in X/G.