# Derivative of vectorized kronecker product

I'm struggling with the following derivative. Let $$\pmb{X}$$ be a symmetrical $$n \times n$$ matrix, $$\pmb{x} = \mathrm{vec}(\pmb{X})$$ and let function $$\pmb{f}$$ take the following form: $$\pmb{f} = \mathrm{vec}\left( \pmb{X} \otimes \pmb{I}_n \right)$$ This is part of a vectorized Jacobian for which I am trying to derive a Hessian. I want to find a solution for: $$\frac{\partial \pmb{f}}{\partial \pmb{x}}$$ I recognized that there is some $$n^4 \times n^2$$ matrix $$\pmb{Y}$$, likely containing only zeroes and ones, such that: $$\mathrm{vec}\left( \pmb{X} \otimes \pmb{I}_n \right) = \pmb{Y} \pmb{x},$$ as $$\pmb{f}$$ just contains only the elements of $$\pmb{x}$$ multiple times including a lot of zeroes. This would automatically imply: $$\frac{\partial \pmb{f}}{\partial \pmb{x}} = \pmb{Y}$$ However, I can't think of a logical structure for $$\pmb{Y}$$.

Let {$$e_k$$} be the standard vector basis, or if you prefer, the columns of $$I_n$$.
Then \eqalign{ M &= \pmatrix{I_n\otimes e_1\cr I_n\otimes e_2\cr \vdots \cr I_n\otimes e_n} \cr \frac{\partial f}{\partial x} = Y &= \Big(I_n\otimes M\Big) \cr } A related problem is \eqalign{ g &= {\rm vec}(I_n\otimes X) \cr \frac{\partial g}{\partial x} &= \Big(M\otimes I_n\Big) \cr }