I  asked this  [question at MSE](http://math.stackexchange.com/questions/1438481/a-question-on-galois-theory) 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?