Homotopy periodicity and coherence

If $f:Z\to Z$ is a periodic homotopy equivalence ( ${\mathrm{id}}_Z\simeq f^p$ ) or a homotopy idempotent ( $f\simeq f^2$ ), the question arises whether this periodicity property can be achieved by a homotopy “compatible with” $f$ . These coherence questions are answered.