I am trying to calculate \begin{align} \frac{\partial\|A(I_n\otimes B)x\|^2}{\partial B}, \end{align} where $\otimes$ denotes the Kronecker product, $A\in\mathbb{R}^{\ell\times mn}$, $B\in\mathbb{R}^{m\times p }$, and $x\in\mathbb{R}^{np}$.
I have expanded the 2-norm and the Kronecker and I calculated the derivatives, but I cannot recollect the terms without exploiting the block structure of $A$ and $B.$
I also tried to use $\mbox{vec}(ABC)=(C^{\rm T}\otimes A)\mbox{vec}B,\;$ but that did not work.
Is there an easy way or any helpful tricks to do this derivative?