Feature importance after PCA (or other dimensionality reduction methods)
Most of the PCA methods return the linear transformation matrix which allows to convert from Component to Variable and viceversa.

What is impossible to do is to assign a feature importance from component to variable (unless you are willing to accept the assumption that "feature importance" could be assigned linearly).

If you accept the idea between parenthesis, you could have a linear equation system $Ax = b$ where $A$ is the component-to-variable matrix (of size components-by-variables), $b$ is the feature importance and $x$ is the result (the variable importance).

This idea I am giving to you is what I think could work, you are likely not going to find it in any research or similar.