All-optical magnetic recording based on the inverse Faraday effect has become a research hotspot in recent years due to its ultra-high storage density and ultra-fast magnetization reversal rate. Existing studies have shown that by optimizing the phase modulation or performing 4π tight focusing of azimuthally polarized vortex beams, high-resolution longitudinal magnetization fields with different axisymmetric intensity patterns can be generated. In order to meet the requirements of more complex all-optical magnetic recording and asymmetric magnetic particle capture and manipulation, it is particularly important to generate an asymmetric light-induced magnetization field with adjustable center position. Studies have shown that the fractional vortex phase could lead to the asymmetric focal field distribution generation under tight focusing conditions, which means that the tightly-focused azimuthally polarized light carrying the fractional vortex phase can produce a novel asymmetric light-induced magnetization field. As a new degree of freedom for the regulation of the magnetization field, the fractional topological charge will bring more new phenomena, new effects and new applications in the field of interaction between light and matter. In this work, for the first time to our knowledge, the magnetization induced by tightly focused azimuthally polarized fractional vortex beam is studied based on the Richard-Wolf vector diffraction theory and the inverse Faraday effect. The equivalent approximation of the magnetization induced by azimuthally polarized fractional vortex beam regarded as a weighted superposition of magnetization induced and crossly induced by a finite number of azimuthally polarized adjacent integer-order vortex beams, where the number of the equivalent terms is chosen by using the histogram intersection method of the intensity distribution image of the magnetization field. The magnetization field distribution under different values of α are also numerically simulated. Studies have shown that magnetization induced by the azimuthally polarized fractional vortex beam is asymmetrically distributed. When the fractional vortex topological charge α belongs to 0.5,1.5, as the vortex topological load increases, the splitting phenomenon of the transverse distribution of magnetization field appears with the magnetization spot position shift in the direction perpendicular to the optical axis. When α equals 0.5 or 1.5, the maximum offset of the center of the magnetization spot is 0.24λ. When the fractional vortex topological charge α belongs to 2,3, the transverse distribution of magnetization field splits two hot intensity spots with gradually growing outer ring diameter. When the fractional vortex topological charge α tends to be an integer 3, the transverse distribution of magnetization field also gets round and symmetrical. In particular, when the fractional vortex topological charge α is a half-integer, especially larger than 3. The number of hot spots (\alpha - 0.5 ) of the intensity of the magnetization field and the number of dark spots ( \alpha - 1.5 ) surrounded by them all have a positive correlation with the number of vortex order. The research in this paper is expected to have new applications in the fields of all-optical magnetic recording and the capture and manipulation of magnetic particles.