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In this paper we study the locally optimal geodesic ball packings with equalballs to the $\mathbf{S}^2\!\times\!\mathbf{R}$ space groups having rotationpoint groups and their generators are screw motions. We determine and visualizethe densest simply transitive geodesic ball arrangements for the above spacegroups, moreover we compute their optimal densities and radii. The densestpacking is derived from the $\mathbf{S}^2\!\times\!\mathbf{R}$ space group$\mathbf{3qe.~I.~3}$ with packing density $\approx 0.7278$.  E. Moln\'ar has shown, that the Thurston geometries have an unifiedinterpretation in the real projective 3-sphere $\mathcal{PS}^3$. In our work weshall use this projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ geometry.

Densest geodesic ball packings to $\mathbf{S}^2\!\times\!\mathbf{R}$ space groups generated by screw motions