Let $G$ be a group and let $H,K$ be subgroups. I have an $S<K$ such that $S=g^{-1}Hg$. I'm asked to show that there is a morphism from $G/H$ to $G/K$. I have a theorem that says that since $H$ and $S$ are conjugates, $G/H$ and $G/S$ are isomorphic as $G$-sets, hence there's a bijective $G$-Set map $$\phi: G/H \xrightarrow{\cong} G/S.$$
I know that for some algebraic objects (groups, rings, vector spaces), if you have an isomorphism between object A and a sub-object $B\subset C$, then the isomorphism becomes a homomorphism when the codomain is enlarged to $C$. Is $\phi$ somehow a $G$-Set morphism into $G/K$ through a similar enlargement? If not, how does the whole $g^{-1}Hg=S<K$ thing come into play when proving the existence of a $G$-Set morphism from $G/H$ to $G/K$?
Since you have $\phi$ (or perhaps more precisely $\phi^{-1}$), the problem appears to be equivalent to the following
Now start by showing that $H g \mapsto K g$ is a well-defined map.
Or if you prefer, first show that the map $\psi : G/S \to G/K$ given by $S g \mapsto K g$ is well-defined, and a morphism of $G$-set, then consider the composition of $\phi^{-1}$ and $\psi$.