Closed function sets on groups of prime order

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 $\mathrm{Clo}({\mathbb{Z}}_p\times {\mathbb{Z}}_p,+)$ for $p>2$.