Motivated by this post (Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?) 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)