Three-space properties of some kinds of coset spaces

In this paper, some three-space properties of (compactly-fibered, neutrally-fibered) coset spaces are considered. Let $X=G/H$ be a coset space and K a closed subgroup of G with $H\subset K$. It is mainly shown that (1) If $G/H$ is neutrally-fibered, then $G/H$ is second-countable ⇔ $K/H$ and $G/K$ are second-countable; (2) If $G/H$ is compactly-fibered such that all compact (resp., countably compact) subspaces of $K/H$ and $G/K$ are metrizable, then all compact (resp., countably compact) subspaces of $G/H$ are metrizable; (3) If $G/H$ is neutrally-fibered such that $K/H$ is second-countable and $G/K$ an ${\aleph }_0$-space (resp., cosmic), then $G/H$ is an ${\aleph }_0$-space (resp., cosmic); (4) If $G/H$ is neutrally-fibered such that $K/H$ is second-countable and $G/K$ has a star-countable cs-network, then $G/H$ has a star-countable cs-network; (5) If $G/H$ is compactly-fibered such that $K/H$ is locally compact metrizable and $G/K$ stratifiable (resp., k-semi-stratifiable), then $G/H$ is stratifiable (resp., k-semi-stratifiable).