I am analyzing psych experiments which generally take the form of a list of outcome measures for the test and control groups, with roughly equal sample sizes, and I would like to compare their means using a permutation test. Most answers suggest this is OK, but this article suggests that Type I errors will be inflated: https://academic.oup.com/bioinformatics/article/22/18/2244/317881.
My understanding from the paper above is that if I am testing for a difference in means using a permutation test, I may get a Type I error/false positive in a situation where the means are equal, but some other feature of the data like variance or skewness differs. In normal life, variance/skewness of data is basically never equal, so I'm having trouble getting an intuitive sense of what this should mean in practice.
Given this, when and how should I conclude that a permutation test is appropriate for testing a difference in means, if ever?