Let's say we have $K \subset H \subset G$, for context, let us identify these objects are groups. We shall also assume, in this case, that $K,H$ are normal in $G$. Then does it make sense to talk about the quotient $K/H$? Similarly, can one talk about $H/K$ if $K$ is only normal in $G$? Because the quotient map $\phi : H \to H/K$ dictates that $\ker \phi = K$, so appears so.
Yet $\pi : H \to K/H$ seems to suggest $\ker \pi = H$ since identity is the same for all 3 groups.
If $K\subset H \subset G$, then no assumptions about normality will help the expression $K/H$ make sense; it just doesn't mean anything.
However, even if $K$ is not normal in $G$, it may still be normal in $H$, in which case you can still form the quotient group $H/K$.
You seem to be a little confused about how quotient homomorphisms work. Given a group $A$ and a normal subgroup $B\trianglelefteq A$, there is a quotient homomorphism $q:A\to A/B$, defined by $q(a)=aB$, and $\ker(q)=B$. Neither of the maps $\phi:K\to H/K$ or $\pi:H\to K$ fit this description, or even really make sense; what is their explicit definition supposed to be.