Some characterizations of $\omega$-balanced topological groups with a $q$-point

In this paper, we study some characterizations of $q$-spaces, strict $q$-spaces and strong $q$-spaces under $\omega$-balanced topological groups as follows: (1) A topological group $G$ is $\omega$-balanced and a $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a countably compact invariant subgroup $H$ which is of countable character in $G$, such that $H\subseteq O$ and the canonical quotient mapping $p:G\to G\mathrm{/}H$ is quasi-perfect and the quotient group $G\mathrm{/}H$ is metrizable. (2) A topological group $G$ is $\omega$-balanced and a strict $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a closed sequentially compact invariant subgroup $H$ which is of countable character in $G$, such that $H\subseteq O$ and the canonical quotient mapping $p:G\to G\mathrm{/}H$ is sequential-perfect and the quotient group $G\mathrm{/}H$ is metrizable. (3) A topological group $G$ is $\omega$-balanced and a strong $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a closed sequentially compact invariant subgroup $H$ of countable character $\{V_n:n\in \omega \}$, such that $H\subseteq O$ and $\{V_n:n\in \omega \}$ is a strong $q$-sequence at each $y\in H$, in $G$ such that the canonical quotient mapping $p:G\to G\mathrm{/}H$ is strongly sequential-perfect and the quotient group $G\mathrm{/}H$ is metrizable.