Let $G_{1}, G_{2}$ be methods on topological spaces $X$ and $Y$ respectively, $f:X\rightarrow Y$ be a mapping, $\mathscr{P}$ be a cover of $X$. $f$ is said to be a $(G_{1}, G_{2})$-quotient mapping provided $f^{-1}(U)$ is $G_{1}$-open in $X$, then $U$ is $G_{2}$-open in $Y$. $\mathscr{P}$ is called a $G$-$cs'$-network of $X$ if whenever $x=\{x_n\}_{n\in\mathbb{N}}\in c_{G}(X)$ and $G(x)=x\in U$ with $U$ open in $X$, then there exists some $n_{0}\in \mathbb{N}$ such that $\{x ,x_{n_{0}}\}\subset P \subset U$ for some $P\in \mathscr{P}$. $\mathscr{P}$ is called a $G$-kernel cover of $X$ if $\{(U)_{G}:U\in \mathscr{P}\}$ is a cover of $X$. In this paper, we introduce the concepts of $(G_{1}, G_{2})$-quotient mappings, $G$-$cs'$-networks and $G$-kernel covers of $X$, and study some characterizations of $(G_{1}, G_{2})$-quotient mappings, $G$-$cs'$-networks, and $G$-kernel covers of $X$. In particular, we obtain that if $G$ is a subsequential method and $X$ is a $G$-Fréchet space with a point-countable $G$-$cs'$-network, then $X$ is a meta-Lindelöf space.