  We classify homomorphisms from mapping class groups by using finite
subgroups. First, we give a new proof of a result of Aramayona--Souto that
homomorphisms between mapping class groups of closed surfaces are trivial for a
range of genera. Second, we show that only finitely many mapping class groups
of closed surfaces have non-trivial homomorphisms into
$\text{Homeo}(\mathbb{S}^n)$ for any $n$. We also prove that every homomorphism
from $\text{Mod}(S_g)$ to $\text{Homeo}(\mathbb{S}^2)$ or
$\text{Homeo}(\mathbb{S}^3)$ is trivial if $g\ge 3$, extending a result of
Franks--Handel.
