  Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables,
and $\eta$ be a counting random variable independent of this sequence. We
consider conditions for $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the
distribution function of the random sum
$S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of consistently
varying distributions. In our consideration, the random variables
$\{\xi_1,\xi_2,\ldots\}$ are not necessarily identically distributed.
