Symmetries of various sets of polynomials

Let K be a field of characteristic 0 and k >= 2 be an integer. We prove that every K-linear bijection f: KX KX preserving the set of k-free polynomials (or the set of polynomials with a k-fold root in K) is a constant multiple of a K-algebra automorphism of KX, i. e. , there are elements a, c in Kᵗimes, b in K such that f (P) (X) = c P (a X + b). When K is a number field or K=R, we prove that similar statements hold when f preserves the set of polynomials with a root in K.