  In this paper, the main aim is to consider the mapping properties of the
maximal or nonlinear commutator for the fractional maximal operator with the
symbols belong to the Lipschitz spaces on variable Lebesgue spaces in the
context of stratified Lie group, with the help of which some new
characterizations to the Lipschitz spaces and nonnegative Lipschitz functions
are obtained in the stratified groups context. Meanwhile, Some equivalent
relations between the Lipschitz norm and the variable Lebesgue norm are also
given.
