Quotient groups of locally graded groups and groups of certain Kurosh-Chernikov classes

We establish the validity of the inclusion G/N∈X for groups G ∈X under certain restrictions on N ⊴ G, where X is one of the following classes, the class of locally graded groups, the class of RI-groups, or the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat P\mathfrak{Y}$$ \end{document} for a fixed group variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{Y} \supseteq \mathfrak{A}$$ \end{document}.