A graph G = (V, E) is called (k, )-sparse if |F | ≤ k|V (F )| − for any nonempty F ⊆ E, where V (F ) denotes the set of vertices incident to F . It is known that the family of the edge sets of (k, )-sparse subgraphs forms the family of independent sets of a matroid, called the (k, )-count matroid of G. In this paper we shall investigate lifts of the (k, )-count matroids by using group labelings on the edge set. By introducing a new notion called near-balancedness, we shall identify a new class of matroids whose independence condition is described as a count condition of the form |F | ≤ k|V (F )| − + α ψ (F ) for some function α ψ determined by a given group labeling ψ on E.