Let $X,Y$ be finite sets, and let $\Sigma(X),\Sigma(Y)$ be their respective permutation groups. Consider a function $f:X\to Y$.
Is there a homomorphism $\phi:\Sigma(X)\to\Sigma(Y)$ induced somehow by $f$?
If $f$ is injective (basically an inclusion), then $\Sigma(X)$ will be isomorphic to a subgroup of $\Sigma(Y)$. But if $f$ is not injective, I guess there are some conditions to verify, right? How does it work?
A reference would also be very welcome.