The documentation states this homomorphism is a permutation equivalence. Here this can be understood as meaning the return value of ActionHomomorphism. But it is easy to make ActionHomomorphism return a map which is not a permutation equivalence.
To make explicit what @ssiccha only hints at: What "permutation equivalence" really refers to the (implicit) permutation equivalence between Sym(Omega) and Sym(|Omega|). That should indeed be clarified.
The documentation states
this homomorphism is a permutation equivalence
. Herethis
can be understood as meaningthe return value of ActionHomomorphism
. But it is easy to makeActionHomomorphism
return a map which is not a permutation equivalence.