Non-fundamental trunc-simplex tilings and their optimal hyperball packings and coverings in hyperbolic space I. For families F1-F4

Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Moln'{a}r et al. in $2006$. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L.~Fejes~T'oth for balls and horoballs in $1964$.