Coset spaces and cardinal invariants

We present an example of a closed subgroup H of a topological group G such that the quotient space G/H is compact, first-countable, and satisfies πw(G/H)<w(G/H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi}w(G/H) < w(G/H)$$\end{document}. We show that if H is a closed neutral subgroup of a topological group G such that G/H has countable character (pseudocharacter), then G/H is metrizable (submetrizable).