Find an example of sets of cosets of different cardinality $G$ is a finite group. Let $H$ be a subgroup of $G$. Is there an example of $G$ and $H$ such that
$${\rm Card}(\{Hxh\mid h\in H\})\neq{\rm Card}(\{Hyh\mid h\in H\}),$$
where $x,y\in G\setminus H$? Here ${\rm Card}$ means cardinality, namely the number of elements contained in a set, so I wonder if we can find two sets of cosets $\{Hxh\mid h\in H\} $ and $\{Hyh\mid h\in H\}$ such that the number of cosets contained in one set is different from the other one.
Could you give me some help? Thank you!
 A: First consider $G=S_3$ and $H=\{e,(1\ 2)\}$, with $x=e$ and $y=(1\ 2\ 3)$. Then
$$
\{Hxh\mid h\in H\} = \{H\} \quad\text{while}\quad \{Hyh\mid h\in H\} = \{H(1\ 2\ 3), H(1\ 2)\}.
$$
But wait, you say, we're not allowed to take $x\in H$? This isn't actually that serious a restriction, since for any nontrivial group $K$ and any $k\in K\setminus\{e\}$, we can now replace $G$ by $G\times K$ and $H$ by $H\times\{e\}$, and $x$ and $y$ by $x\times k$ and $y\times k$.
A: Consider the symmetry group $D_8$ of a square.  Let $C_2$ denote the subgroup fixing a corner $x$ (so $C_2$ consists of the identity, and reflection through the diagonal containing $x$).  Then we can identify the $4$ corners of the square with the cosets of $C_2$. That is all the elements of $C_2g$ map $x$ to $xg$, so we may identify the coset $C_2g$ with the corner $xg$, for each $g\in D_8$.
The orbits of corners under $C_2$ have different sizes: one orbit is the two corners adjacent to $x$, another is the single corner opposite to $x$.
Thus if $a$ is a $90^\circ$ rotation, then $\{C_2ah|h\in C_2\}$ is two cosets, whilst $\{C_2a^2h|h\in C_2\}$ is one.
