Is there an action of the abelianization of $SL(2,\mathbb{Z})$ on lattices?

Question

Since there is an action of the group $\text{SL}(2,\mathbb{Z})$ on the lattice $\mathbb{Z}^2$, and knowing that its abelianization is the group $\mathbb{Z}_{12}$, is there a corresponding quotient action of this cyclic group on lattices as well ?