Remarks

Another way of describing a permutation group is as a subgroup of Sym(X) generated by a set of permutations, i.e., it is a a set of permutations closed under multiplication and taking inverses, and containing the identity element.

If f: H -> Sym(X) is injective, then f determines an isomorphism of H on the subgroup f(H) of Sym(X).
Identifying H with its image under f, the group H itself is sometimes also called a permutation group on X.