Symmetries of various sets of polynomials

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly 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., that there are elements $a, c \in K^{\times}$ and $b \in K$ such that $f(P)(X) = c P(a X + b)$. When $K$ is a number field or $K=\mathbb R$, we prove that similar statements hold when $f$ preserves the set of polynomials with a root in $K$.