Let $X$ be a topological space on which a group $G$ acts . let $N$ and $K$ be subgroups of $G$. under what condition we have an induced action of $K$ on $X/N$?
My guess: if $N$ is normalized by $K$ in $G$ then we have an induced action. Indeed, we have an action $$K\times X/N\rightarrow X/N;\; (k,[x])\mapsto [kx]$$ This is a well defined action since $[x]$ and $[nx]$ map to the same image $[kx]$ This is because $\forall n\in N$ and $\forall k\in K$ and $\forall x\in X$, $knx=(knk^{-1})kx=n'kx$ where $n'= (knk^{-1})\in N$ since $N$ is normalized by $K$, hence $[knx]=[kx]$.