  We give a full description of all sets of functions on the group $(\mathbb{
Z}_p, +)$ of prime order which are closed under the composition with the clone
generated by $+$ from both sides. Thereby, we also get a description of all
iterative algebras on $\mathbb{Z}_p$ which are closed under the composition
with the clone generated by $+$ from both sides. As another consequence, there
are infinitely many non finitely generated clones above
$\operatorname{Clo}({\mathbb{ Z}_p \times \mathbb{ Z}_p, +})$ for $p > 2$.
