Is it possible to factor this matrix
$$\begin{bmatrix} x_{11} a_{11} & x_{11} a_{12} & x_{12} a_{11} & x_{12} a_{12} & \\ x_{21} a_{21} & x_{21} a_{22} & x_{22} a_{21} & x_{22} a_{22} & \\ \end{bmatrix}$$
as a function of
$$ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} $$
or $\begin{bmatrix} A &|& A\end{bmatrix}$, possibly without using the Kronecker product?
PS: This is just a simplified $2 \times 2$ version, the dimensions are actually arbitrary.
I think this is not possible. But at least this is possible:
\begin{bmatrix} \mathrm{diag}(x_{11},x_{21})\cdot A & \mathrm{diag}(x_{12},x_{22})\cdot A \end{bmatrix}