Random convolution of inhomogeneous distributions with $\mathcal{O}$-exponential tail

Let $\{{\xi }_1,{\xi }_2,\dots \}$ be a sequence of independent random variables (not necessarily identically distributed), and $\eta$ be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{{\xi }_1,{\xi }_2,\dots \}$ 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 $\mathcal{O}$-exponential distributions.