Free groups in the smallest ideal of βG

Let G be an infinite group with cardinality κ embeddable into a direct sum of countable groups and let βG be the Stone-Čech compactification of G as a discrete semigroup. We show that the structural group of the smallest ideal of βG contains copies of the free group on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{2^{\kappa}}$\end{document} generators.