Assume that a group $G$ has a presentation $\langle X \mid R \rangle$ and a group $H$ has a presentation $\langle X \mid S \rangle$. If $R \subseteq S$, the $H$ is isomorphic to a quotient of $G$.
In fact, if $F_X$ is the free group and $\varphi: F_X \rightarrow G$ and $\psi:F_X \rightarrow H$ are the homomorphism, then $\ker \varphi = R^{F_X} \leq S^{F_X} = \ker \psi$. So, $H \cong F_X/\ker\psi$ is isomorphic to a quotient of $F_X/\ker\varphi \cong G$.