  We study abelian-by-cyclic Moufang loops. We construct all split
$3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang
permutations on abelian groups $(X,+)$, which are permutations that deviate
from an automorphism of $(X,+)$ by an alternating biadditive mapping
(satisfying certain properties). More generally, we obtain additional
abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside,
we show that in the Moufang loops $Q$ obtained from a construction pair on
$(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of
$Q$ if and only if $Q$ is a group.
