Ni$_{1-x}$(Zn$_{0.6}$Mg$_{0.2}$Cu$_{0.2})_x$ Fe$_2$O$_4$ (where x = 0.0, 0.3, 0.5 and 0.7) was prepared by the auto combustion method and was characterized by X-ray diffraction (XRD). From XRD analysis, the average crystallite size and lattice constant were determined in the range of 6–11 nm and 8.331–8.372 Aº, respectively. The structure of single-phase cubic in samples was also confirmed because the main peaks of the patterns were related to the position of the peaks in the mentioned standard (JCPDS card no. 00-008-0234). The average size of the particle was calculated in the range of 21–26 nm from the field emission scanning electron microscopy (FESEM). The atomic percentage of elements was obtained from the energy dispersive spectroscopy (EDS). From room-temperature hysteresis loops, the magnetization was evaluated in a magnetic field of 9 kOe. It was observed that the magnetization was not saturated and was increased with the increment in x values. According to the core–shell model, the magnetic behaviour of samples were discussed. The results illustrated that the magnetization of the sample with x = 0.7, due to the larger particle size than that of other samples, was the highest. In addition, the increase in magnetization was also discussed based on Neel’s theory and the site’s occupation of the substituted cations on the sub-lattices. Also, the variation of coercivity was investigated. The coercivity, due to the multidomain structure of samples, was decreased with doping Zn, Mg and Cu. Among all the samples, the sample with x = 0.7 indicated the lowest coercivity and the sample with x = 0 showed the highest coercivity. Based on the domain wall movement, the variations of initial permeability as a function of frequency in the frequency range of 10 kHz–10 MHz were discussed. The results indicated that the initial permeability of the sample with x = 0.7, similar to its magnetization, was the highest. The constancy of ${\mu}'$ in the frequency range of 10 kHz–1 MHz was due to the motion of the domain wall, which was indicated by the compositional stability and quality of the samples. Finally, the results obtained from the initial permeability in the present work were compared to the values obtained from the previous researchers. Generally, it was observed that the sample with x = 0.7, due to the maximum magnetization, highest initial permeability and lowest coercivity could be suitable for high-frequency applications.