1. Use Gauss's law

   Gauss's law states: $\Phi = \frac{q_{\text{enclosed}}}{\varepsilon_0}$ where $\Phi$ is the total electric flux through a closed surface.

2. Understand the geometry

   The charge $q$ is placed at the center of one face of the cube, i.e. on the boundary of the cube.

   To handle such cases, imagine placing an identical cube adjacent to the given cube, sharing that face. Then the charge lies exactly at the common face and becomes the center of the resulting rectangular box made of 2 identical cubes.

3. Apply symmetry

   The combined closed surface (the rectangular box of 2 cubes) encloses the charge $q$ at its center.

   Hence, total flux through the combined surface is: $\Phi_{\text{total}} = \frac{q}{\varepsilon_0}$

   By symmetry, the flux is equally divided between the two identical cubes.

   Therefore, flux through one cube is: $\Phi_{\text{cube}} = \frac{1}{2}\cdot \frac{q}{\varepsilon_0} = \frac{q}{2\varepsilon_0}$

4. Check options

   • A: $\frac{q}{2\varepsilon_0}$ ✅
   • B: $0$ ❌
   • C: $\frac{q}{4\varepsilon_0}$ ❌
   • D: $\frac{q}{8\varepsilon_0}$ ❌

5. Final answer

   $\boxed{\frac{q}{2\varepsilon_0}}$

So the correct option is A.