Geometric progression stabilizer in common metric

So-called normalized metrics are considered on the set of elements of a geometric progression. A full description of normalized metrics with maximal stabilizer, which is the group of integer degrees of the common ratio of the progression, is presented. Previously, it was known that this group is the stabilizer for the minimal normalized metric (inherited from the real line) and the maximal normalized metric (an intrinsic metric all paths in which pass through zero). The stabilizer of a metric space is understood as the set of positive numbers such that multiplying the metric by this number produces a metric space lying at a finite Gromov-Hausdorff distance from the original space.