Do we permute columns in the test set when running Permutation Importance?

Question

I've been looking at the documentation and relevant tutorials on permutation importance and nobody seems to have a clear idea on what they are actually permuting.

Just to clarify, would the step by step process be as follows:

1. Split dataset into X_train, X_val and X_test

2. Train data on X_train, using X_val to e.g. find best epoch

3. Run the trained model on X_test, taking note of the metric we are measuring

4. Permuting a feature in X_test, and running the same model on this permuted X_test dataset

5. Take note of the same metric and compare the two

6. Repeat for each variable, without changing the model.

Aside Question: Would it be worth running repeats of this permutation process, with X_train,X_val and X_test changing with each repeat. I know the resultant model will be different, but I want to get a broad perspective of how the general model (with fixed hyperparameters) behaves when trained on different datasets, as keeping X_test fixed may skew the perceived importances of certain features.