  T\u{a}rn\u{a}uceanu described the finite groups $G$ having exactly $|G|-1$
cyclic subgroups. In "Finite Groups with a Prescribed Number of Cyclic
Subgroups,", we used elementary methods to completely characterize those finite
groups $G$ having exactly $|G|-\Delta$ cyclic subgroups for $\Delta=2, 3, 4$
and $5$. In this paper, we prove that for any $\Delta >0$ if $G$ has exactly
$|G|-\Delta$ cyclic subgroups, then $|G|\le 8\Delta$ and therefore the number
of such $G$ is finite. We then use the computer program GAP to find all $G$
with exactly $|G|-\Delta$ cyclic subgroups for $\Delta=1,\ldots,32$.
