Rewrite matrix operation involving diagonal matrix and Kronecker product

Question

$D$ is a diagonal matrix in $\mathbb{R}^{(kn) \times (kn)}$, $W$ is in $\mathbb{R}^{n \times m}$. Does there exist a matrix $A \in \mathbb{R}^{n \times m}$ so that $D (W \otimes I_{k}) = A \otimes I_{k}$?

Thanks for the help.

Answer 1

Ok, so as you write in the comments, you know that there are some $D$ which cannot be written as a Kronecker product. Now if we take $m = n$ and $W = I_n$, we'll get a counterexample.