For the one-sample $t$-test a new form of the exact distribution of the test statistic $t^2$ is obtained when sampling from a distribution which is a mixture of two normal distributions. A numerical example is provided to show that the size of the test can differ greatly when sampling from distributions having the same skewness and kurtosis. Contours of equal size are plotted for a particular case in a certain cross section of the parameter space.
"One-Sample $t$-Test When Sampling from a Mixture of Normal Distributions." Ann. Statist. 5 (4) 803 - 807, July, 1977. https://doi.org/10.1214/aos/1176343904