Quotient group, group action and quotient space

Question

Let $G$ be a group acting faithfully on a topological space $X$, and $N$ a normal and abelian subgroup of $G$.
Does this mean that one can define a faithful action of the quotient group $G/N$ on the quotient space $X/N$?

Answer 1

Consider the discrete space $X=\{1,2,3\}$. $S_3$ acts faithfully on $X$ by definition. $A_3 \subset S_3$ is an abelian normal subgroup, but it acts transitively, so $X/A_3$ has only one point. Thus the quotient group $S_3/A_3$ cannot act faithfully on $X/A_3$.