In my opinion, it depends on the degree in which the assumption of normality is violated. The Fisher ANOVA is known to be quite robust. Inspect the histograms and QQ-pots. The Shapiro test is known to be biased by size. I would choose the Kruskal-Wallis only if the data are heavily skewed. In case of homogeneity of variances I would not choose the Welch ANOVA.

See also attachment.

Don't forget that there are robust versions of these things. You might want to look at the work of Rand Wilcox at the University of Southern California. I find his work very helpful. Best regards, David Booth

In the case of homoscedasticity, results of Welch's t test will be same as those from Student's t test.

Also note, in the general case, Kruskal-Wallis doesn't assume homoscedasticity. Other than independence of observations and having at least ordinal data, Conover, 1999, Practical Nonparametric Statistics, 3rd, has the assumption for the test as "Either the k population distribution functions are identical, or else some of the populations tend to yield larger values than other populations do." That's it. I think it's unfortunate that some authors present a version of the test that assumes something like that all populations are the same (in shape and spread) except for their location. Under these assumptions, the test can be used as a test of location (mean or median). But it seems silly to me to add distributional assumptions to a nonparametric test. The natural hypotheses of the KW test --- dealing with stochastic equality or dominance ---- is often of interest.

Of course there are many ways to compare two groups, depending on what hypothesis you want to test (Do you care about means, medians, stochastic dominance, the 75th percentile?), and the kind of data you're dealing with.