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 : K[X] → K[X] 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 $K[X]$, 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.