Injectivity of Quadratic Matrix Function

Fix positive integers $D,N$ and $d$.

Let $A$ and $W$ be $N\times N$ and $d\times d$ matrices respectively. Consider the map$$\begin{aligned}f:\mathbb{R}^{N\times d} & \to \mathbb{R}^{N\times d}\\X& \mapsto AXW + X.\end{aligned}$$What conditions do we need on $A$ and on $W$ to guarantee that $f$ is injective?

So far what I have is: If $W$ is the identity matrix then we only want that $(A+I_{N\times d})$ has full column rank...but I have no idea how to handle the general case (ie when W is general)