  In this paper, we generalize the notion of cyclic metric to homogeneous
Finsler geometry. Firstly, we prove that a homogeneous Finsler space $(G/H, F)$
must be symmetric when it satisfies the naturally reductive and cyclic
conditions simultaneously. Then we prove that a Finsler cyclic Lie group which
is either flat or nilpotent must have an Abelian Lie algebra. Finally, we show
how to induce a cyclic $(\alpha,\beta)$ metric from a cyclic Riemannian metric.
Using this method, we construct a Randers cyclic Lie group.
