# Does each weakly feathered topological group admit an injective homomorphism into a feathered topological group?

A topological group $G$ is called

• $\omega$-$\mathit{narrow}$ if for each non-empty open set $U\subset G$ there exists a countable subset $C\subset G$ such that $G=CU=UC$;
• $\mathit{feathered}$ if $G$ contains a compact subset $K\subset G$ having countable neighborhood base in $G$;
• $\mathit{weakly\ feathered}$ if $G$ contains a compact $G_\delta$-set.

It can be shown that a topological group $G$ is (weakly) feathered if and only if $G$ contains a compact subgroup $K$ such that the quotient space $G/K$ is (sub)metrizable.

Problem. Does each weakly feathered $\omega$-narrow topological group admit a continuous injective homomorphism to a feathered topological group?

Remark. It can be shown that each $\omega$-narrow topological group of countable pseudocharacter admits a bijective continuous homomorphism onto a metrizable separable topological group.