A model of the induced gravity is proposed that, under a conformal transformation, becomes a chiral cosmological model with two fields. To form primordial black holes, a second inflationary phase is necessary, which occurs near the quasi-de Sitter solution. The length of the plateau in the Hubble parameter of the Einstein system determines the mass of the primordial black holes. In models with two fields, there is initially a slow roll along the classical inflaton field, followed by a second field that grows. The Hubble parameter during the second phase has a sufficient number of e-folds to form primordial black holes during the radiation-dominated era, with masses similar to dark matter candidates. To analyze single-field models of induced gravity that correspond to the second inflation phase near the quasi-de Sitter point, the superpotential method was developed.