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