  Sharifi has constructed a map from the first homology of the modular curve
$X_1(M)$ to the $K$-group $K_2(\mathbf{Z}[\zeta_M, \frac{1}{M}])$, where
$\zeta_M$ is a primitive $M$th root of unity. We study how these maps relate
when $M$ varies. Our method relies on the techniques developed by Sharifi and
Venkatesh.
