  Let $\Gamma$ be either the mapping class group of a closed surface of genus
$\geq 2$, or the automorphism group of a free group of rank $\geq 3$. Given any
homological representation $\rho$ of $\Gamma$ corresponding to a finite cover,
and any term $\mathcal{I}_k$ of the Johnson filtration, we show that
$\rho(\mathcal{I}_k)$ has finite index in $\rho(\mathcal{I})$, the Torelli
subgroup of $\Gamma$. Since $[\mathcal{I}: \mathcal{I}_k] = \infty$ for $k >
1$, this implies for instance that no such representation is faithful.
