The fractal is a kind of geometric figure with self-similar character. Phase transition and critical phenomenon of spin model on fractal lattice have been widely studied and many interesting results have been obtained. The \begin{document}${S^4}$\end{document} model regarded as an extension of the Ising model, can take a continuous spin value. Research of the \begin{document}${S^4}$\end{document} model can give a better understanding of the phase transition in the real ferromagnetic system in nature. In previous work, the phase transition of the \begin{document}${S^4}$\end{document} model on the translation symmetry lattice has been studied with the momentum space renormalization group technique. It is found that the number of the fixed points is related to the space dimensionality. In this paper, we generate a family of diamond hierarchical lattices. The lattice is a typical inhomogenous fractal with self-similar character, whose fractal dimensionality and the order of ramification are \begin{document}${d_{\rm{f}}} = {\rm{1}} + \ln m/\ln {\rm{3}}$\end{document} and \begin{document}$R = \infty $\end{document}, respectively. In order to discuss the phase transition of the \begin{document}${S^4}$\end{document} model on the lattice, we assume that the Gaussian distribution constant \begin{document}${b_i}$\end{document} and the fourth-order interaction parameter \begin{document}${u_i}$\end{document} depend on the coordination number \begin{document}${q_i}$\end{document}