This paper studies the period map of the family \(\{{\mathcal{X}}_t\}\), \(t=(t_0,t_1)\in{\mathbb{P}}^1\), of the quartic family of \(K3\) surfaces defined by \[ {\mathcal{X}}_t: x_1+\cdots+x_5= t_0(x_1^4+\cdots+x_5^4)+t_1(x_1^2+\cdots+x_5^2)^2=0 \] in \({\mathbb{P}}^4\) with homogeneous coordinates \((x_1:\cdots :x_5)\). This family admits a symplectic group action by the symmetric group \(S_5\), and in fact it is a maximal family of algebraic \(K3\) surfaces with an action of \(S_5\). The Picard number of a generic fiber is equal to \(19\), and the Gram matrix of the transcendental lattice \(T\) is given by \(\left( \begin{matrix} 4 & 1 & 0 \cr 1 & 4 & 0 \cr 0 & 0 & -20 \end{matrix} \right)\). The image of the period map of this family is a \(1\)-dimensional subdomain \(\Omega_T\) of the \(19\)-dimensional period domain (the bounded symmetric domain of type IV). Let \(\Omega_T^{\circ}\) be a connected component of \(\Omega_T\). Since \(O(T)\) has no cusp, there is a modular embedding \(i : \Omega_T^{\circ}\to {\mathbb{H}}_2\) to the Siegel upper half-plane \({\mathbb{H}}_2\) of genus \(2\). This modular embedding is constructed using the Kuga-Satake construction. The main result of this paper is to construct the inverse of the period map using automorphic forms of one variable on \(\Omega_T^{\circ}\). In fact, automorphic forms are constructed as the pull-backs of the forth power of theta constants of genus \(2\). Also relations among these autormorphic forms are found to obtain a uniformizing parameter of the Shimura curve \(O(T)\setminus \Omega_T\).