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In this paper, the main aim is to consider the mapping properties of themaximal or nonlinear commutator for the fractional maximal operator with thesymbols belong to the Lipschitz spaces on variable Lebesgue spaces in thecontext of stratified Lie group, with the help of which some newcharacterizations to the Lipschitz spaces and nonnegative Lipschitz functionsare obtained in the stratified groups context. Meanwhile, Some equivalentrelations between the Lipschitz norm and the variable Lebesgue norm are alsogiven.

Necessary and sufficient conditions for boundedness of commutators of fractional maximal function in variable Lebesgue spaces on stratified groups