Let $G$ be a group and let $H$, $K$ be normal subgroups in $G$ such that $H<K$. Suppose that $$G=Hx_{1} \dot{\cup} Hx_{2} \dot{\cup} \dots \dot{\cup} Hx_{n}.$$ That is, $\{x_{1},\dots,x_{n}\}$ are coset representatives.
Since $H$ is also a subgroup in $K$, we can talk about the right cosets of $H$ in $K$.
I would like to know if there is a way to know which are the representatives.
I have tried the following:
Let $x_{1},\dots,x_{k}$ be the elements that are in $K$, and $x_{k+1},\dots,x_{n}$ the ones that are in $G$ but not in $K$.
Then, $$K= Hx_{1}\dot{\cup}Hx_{2}\dot{\cup}\dots Hx_{k}.$$
It is trivial that $Hx_{i}\subseteq K$ because $x_{i}$ is in $K$ and $H$ is a subgroup of $K$. On the contrary, if $g\in K$, $g\in G$, so $g\in Hx_{j}$ for some $j\in \{1,\dots,n\}$. Thus, $$Hg=Hx_{j},$$ so $x_{j}\in Hg\subseteq K$.
Is this right? Thanks in advance!