  For any positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of
the mapping class group of a surface of genus at most two such that $\Gamma_n$
admits an epimorphism onto a free group of rank $n$. We conclude that
$H^1(\Gamma_n;\Z)$ has rank at least $n$ and the dimension of the second
bounded cohomology of each of these mapping class groups is the cardinality of
the continuum. In the case of genus two, the groups $\Gamma_n$ can be chosen
not to contain the Torelli group. Similarly for hyperelliptic mapping class
groups. We also exhibit an automorphism of a subgroup of finite index in the
mapping class group of a sphere with four punctures (or a torus) such that it
is not the restriction of an endomorphism of the whole group.
