I am basically trying to solve for $M \in \mathbb{R}^{m \times n}$ below, such that for any vector $\mathbf{x} \in \mathbb{R}^{n^2}$ and some fixed $C \in \mathbb{R}^{m^2 \times n^2}$:
$$ (M \otimes M) \mathbf{x} = C \mathbf{x}$$
If there is a unique solution, I guess this is the same as finding $M$ such that $(M \otimes M) = C$.
Are there conditions under which there are solutions? Any knowledge about this problem would be useful, as I'm not too familiar with it; does it have a name? Are there certain fields where solving for M like this comes up?
Edit: first version of the question incorrectly stated C was square and symmetric.