On the Galois spectra of polynomials with integral parameters

We prove that there exists a polynomial $F(x,t)$ with rational coefficients whose degree with respect to $x$ is equal to 4, such that for every integer the Galois group of the decomposition field of the polynomial $F(x,a)$ is not the dihedral group, but any other transitive subgroup of the group $S_4$ can be represented as the Galois group of the decomposition field of the polynomial $F(x,a)$ for some integer $a$.