  Rational transformations of polynomials are extensively studied in the
context of finite fields, especially for the construction of irreducible
polynomials. In this paper, we consider the factorization of rational
transformations with (normalized) generators of the field $K(x)^G$ of
$G$-invariant rational functions for $G$ a finite subgroup of
$\operatorname{PGL}_2(K)$, where $K$ is an arbitrary field. Our main theorem
shows that the factorization is related to a well-known group action of $G$ on
a subset of monic polynomials. With this, we are able to extend a result by
Lucas Reis for $G$-invariant irreducible polynomials. Additionally, some new
results about the number of irreducible factors of rational transformations for
$Q$ a generator of $\mathbb{F}_q(x)^G$ are given when $G$ is non-cyclic.
