Motivated by this  post (http://mathoverflow.net/questions/219076/a-question-in-galois-theory) we ask the following question:

Assume that $T$ is  a  a linear map over $\mathbb{Q}[x]$, the space  of  polynomials with rational coefficients,    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? (A  non identity  functor)