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In this paper we define the pullback $cat^1$-group and show that this Pullback has a right adjoint which is the induced $cat^1$-group. Later we show that this right adjoint is a pushout of category of cat1-groups. We calculate the Peiffer subgroups to find a finite group of the source of induced cat1-groups. The generating set of Peiffer subgroups are also given in this paper. All results are corrected by a GAP program package in . This paper also contains the some computational examples which are the calculation-induced cat1-group and comparative times between the induced crossed modules and induced cat1-groups.