Q: Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$ I thank  J.C. and Arturo Magidin for interesting suggestions in the comments to this question
In this  post the field of rational numbers  is  denoted by $\mathbb{Q}$. The space  of  polynomials with rational coefficients is  denoted by $\mathbb{Q}[x]$. The commutator  subgroup of a  group $G$ is  denoted  by $G'$. The isomorphism between two groups is  denoted by $\simeq$
We learn from the answer to the following question 

Is there a $\mathbb{Q}$-linear map $T$ on $\mathbb{Q}[x]$ such that for all polynomials $\mathrm{Gal}(T(f))$ is the commutator subgroup of $\mathrm{Gal}(f)$?

that there does not exist a $\mathbb{Q}$-linear map $T$ on $\mathbb{Q}[x]$  such that  for all $f\in \mathbb{Q}[x]$ we have $$Gal(T(f))\simeq(Gal(f))'$$
Motivated by the above fact we ask the following question:

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

What, if any, are some  non trivial examples of such situation? (A  non identity  functor?Some interesting examples?)
