Finite Groups with a Prescribed Number of Cyclic Subgroups

Marius T\u{a}rn\u{a}uceanu described the finite groups $G$ having $\mathrm{\mid }G\mathrm{\mid }-1$ cyclic subgroups. We describe the finite groups $G$ having $\mathrm{\mid }G\mathrm{\mid }-\mathrm{\Delta }$ cyclic subgroups for $\mathrm{\Delta }=2,3,4$ and $5$.