<p>The paper investigates the randomly stopped sums. Primary random variables are supposed to be nonnegative, independent, and identically distributed, whereas the stopping moment is supposed to be a nonnegative, integer-valued, and nondegenerate at zero random variable, independent of primary random variables. We find the conditions under which dominated variation or extended regularity of randomly stopped sums determines the stopping moment to belong to the class of dominatedly varying distributions. In the case of extended regularity, we derive the asymptotic inequalities for the ratio of tails of the distributions of randomly stopped sums and a stopping moment. The results generalize analogous statements recently obtained for a narrower class of regularly varying distributions. Compared with the previous studies, we apply new methods to the proofs of the main statements because methods applied to regularly varying functions are unsuitable for the class of generalized regularly varying distributions. At the end of the paper, we provide one example that illustrates the theoretical results.</p>