We exhibit several finite groups that are not active sums
of cyclic subgroups. We show that this is the case for groups
with $H_{1}G$ of odd order and $H_{2}G$ of even order. As
particular examples of this we have the alternating groups
$A_n$ for $n\geq 4$, some special and some projective linear
groups. Our next set of examples consists of $p$-groups where
the normalizer and the centralizer of every element coincide.
We also have an example of a 2-group where the above conditions
are not satisfied; thus we had to devise an ad hoc argument.
We observe that the examples of $p$-groups given also provide
groups that are not molecular.