On symmetrical $q$-extensions of the 2-dimensional grid

In the paper, properties of symmetrical $q$-extensions of the grids are investigated. We obtain a criteria for sets of symmetrical $q$-extensions of the grid $\Lambda^2$ to be finite. Using this criteria we prove, in particular, that the set of all $Aut_0(\Lambda^2)$-symmetrical $q$-extensions of the grid $\Lambda^2$ is finite for any prime $q$. In addition, we give a list of all $Aut_0(\Lambda^2)$-symmetrical 3-extensions of the grid $\Lambda^2$.