Does variance 'change' when we compare 2 groups?

Yes [and no], depending on the model we believe in. See e.g. below:

1.A 2 sample t-test compares the difference between means against zero

2.ANOVA uses an F test to compare within-groups sums-of-squares (pretty much how far away individual scores are from each group’s mean) with between-group sums-of-squares (how far away group means are from the overall/grand mean)

3.SEM can compare the means using a chi-square test for the 2-group model

ØALL these 3 methods allow for variances different or assumed/set equal; the last method only estimates [from the 'equal variances model', which may fit better/worse than the 'variances different' model] a new (common, equal)  population variance in addition to the 2 population means, i.e. the model yields 1 'NEW' variance, that may differ little/a lot from the 2 sample ones. These differences are taken into account and ‘reported’ along with differences between sample means and estimated means, as part of the FIT of the model. Good fit=ok, bad fit=not ok.