I recently saw a question in a Text book, which asks to prove that "The group of symmetries of the polynomial $x_1x_2 + x_3x_4 + x_5x_6$ is a subgroup of $S_6$ of order $48$".
(By the group of Symmetries of this polynomial, we mean the stabilizer of the polynomial $x_1x_2+x_3x_4+x_5x_6$ in the action of the group $S_6$ on $\mathbb{Z}[x_1,x_2,\cdots,x_6]$ given by $\sigma.f(x_1,x_2,\cdots,x_6)=f(x_{\sigma(1)},x_{\sigma(1)},\cdots,x_{\sigma(6)})$.)
In view of this exercise, I would like to ask the following question:
Is every finite group realizable as the full stabilizer of a polynomial over $\mathbb{Z}$ in a certain number of indeterminates? If yes, then how can we construct that polynomial?