Molecular electronics is a promising candidate to meet the increasing demands of electronic component miniaturization. Within this field, the interaction between a molecular functional center and its environment plays a profound role both in designing molecular devices and in understanding the structure-property relationships at the molecular/atomic level. Herein, motivated by an isomer of the nickel tetraphenylporphyrin (Ni-TPP) molecule, we study the dynamical and thermodynamical properties of such a structure, with the aid of a side-coupled quadruple impurity Anderson model. We demonstrate that by adjusting the central-ligand hopping integral, the magnetic moments of the system are screened in two stages, each accompanied by the rising and falling of the Kondo resonant peak in the transport properties, respectively. However, one can hardly consider this behavior as the two-stage Kondo effect, because the second process is not the result of the screening of the side spins by the electrodes, instead, they are totally shielded by the central spins. Furthermore, if one sweeps the on-site Coulomb repulsion in the ligand orbitals, the magnetic moments are then screened in three stages. We note that the first and third screening processes are accompanied by the appearance and disappearance of the Kondo behavior in the linear conductance, respectively, while the second screening has no influence on the quantum transport through the device. These characteristics are apparently different from the typical three-stage Kondo physics reported by some previous studies. To implement the above results, we adopt the state-of-the-art numerical renormalization group method, combined with numerical simulations and analytical arguments. locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon locked icon Physics Subject Headings (PhySH)Kondo effectMolecular junctionsMolecular magnetsQuantum many-body systemsAnderson impurity modelNumerical Renormalization Group