Equivariant polynomial maps and gradients of invariants

Question

Let $G$ be a finite group with a linear action on $\mathbb{C}^n$ and $f\in\mathbb{C}[X_1,\ldots,X_n]$ be invariant. Then the gradient of $f$ gives rise to a polynomial map $\phi$ from $\mathbb{C}^n$ to $\mathbb{C}^n$. This map is $G-$equivariant, i.e., $g(\phi(x))=\phi(g(x))$ for all $g\in G$. Now I wonder, if I get all $G-$equivariant maps from $\mathbb{C}^n$ to $\mathbb{C}^n$ in this way? (I think this is true for finite reflection groups, but also more generally?)