I asked this question at MSE but I did not received an answer. So I ask it at MO:

We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by $Gal(f)$. The commutator subgroup of a group $G$ is denoted by $G'$.

Is there a $\mathbb{Q}$- linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $f\in \mathbb{Q}[x]$ we have $$Gal(T(f))=(Gal(f))'$$

More generally we can consider the following situation;

Assume that $T$ is a a linear map over $\mathbb{Q}[x]$ and $\mathcal{F}$ is a functor over the category of groups. we say that $T$ is Galois-related to $\mathcal{F}$ if we have $$Gal(T(f))=\mathcal{F}(Gal(f))\;\;\;\text{For all polynomial } f \in \mathbb{Q}[x]$$

What are some non trivial examples of such situation?