Has this "pseudo-quotient" of groups been studied before? Let $G$ be a (finite) group with subgroup $H$. Pick out (left) coset representatives: $x_1=1, x_2, \dots, x_\ell$ (so $x_1H=1H=H$). Now form an $\ell \times \ell$ table with rows and columns labeled by $H, x_2H, \dots, x_\ell H$. 
In addition group cosets together according to their double cosets.
For example, if $Hx_2H=x_2H \cup x_3H \cup x_4H$ we group cosets 2, 3, and 4 together, so we would have something like $(H)$ $(x_2H \; x_3H \; x_4H)$ $(x_5H \cdots$
This (parenthesized) list forms the first row of our table.
The rest of the rows are formed by left multiplying by each representative in turn. So, for example, 
if $Hx_2H=x_2H \cup x_3H \cup x_4H$, we would have
$$\begin{array}{rcccccc}
\mbox{Row }x_1H: & \quad & (H)    & (x_2H    & x_3H    & x_4H)    & (x_5H    & \cdots \\\\
\mbox{Row }x_2H: & \quad & (x_2H) & (x_2x_2H & x_2x_3H & x_2x_4H) & (x_2x_5H & \cdots \\\\
\mbox{etc.} \end{array}$$
Notice that in row $xH$, two cosets $yH$ and $zH$ lie in the same parenthesized group if and only if $x^{-1}yH$ and $x^{-1}zH$ lie in the same double coset. Put another way, cosets $yH$ and $zH$ are connected in row $xH$ if and only if $xHx^{-1}yH = xHx^{-1}zH$. So the parentheses in row $xH$ are recording something about the action of $xHx^{-1}$ on the left cosets of $H$. 
Of course if $H$ is a normal subgroup, this is just the Cayley table for the quotient group $G/H$ (with some extra parentheses). 
When $H$ is not normal, switching representatives necessarily changes the order in which cosets appear in rows 2+ but the parenthesized groups stay intact. This pseudo-quotient gets as close to a group structure as one can hope for when $H$ fails to be normal.
My question, does this look familiar to anyone? References?
Now for some much needed explanation. About one year ago I got a call out of the blue by someone living close to my university who said he had a short proof of the Feit-Thompson odd order theorem. 
I was quite skeptical but told him to email me his proof, I'd look it over and we could talk. It turns out he is a retired mathematician and who has worked off-and-on on his proof for 30+ years as a hobby. While I still have serious doubts his "proof" can be adequately formalized and/or corrected, I did find his fundamental object of study interesting -- these "pseudo-quotients" which he call his "diagrams". 
These pseudo-quotients allow one to almost quotient groups even when there are no normal subgroups around. Using these pseudo-quotients, I thought we had a character free proof that the Frobenius Kernel is a normal subgroup, but it fell apart and I haven't been able to repair the proof.
While these quotients haven't helped me prove anything nontrivial (yet), they have caused me to "trip over" a lot of old important theorems.
 A: This is very interesting. I don't really understand what is going on with the double cosets, but this is reminiscent of the standard construction of loops (quasigroups with identity elements) as tranversals in groups. This dates back to Baer in the early 40's. The idea is the following. You have a group $G$, a subgroup $H$, and a transveral $T$, that is a set of coset representatives. Assume further that $T$ is normalized, that is, $1\in T$. For $x,y \in T$, define an operation $\circ$ on $T$ by $(x\circ y)H = xyH$, that is, $x\circ y$ is the unique representative in $T$ of $xyH$. It is easy to see that $1$ is the identity element for $\circ$ and that for all $a,b\in T$, the equation $a\circ x = b$ is uniquely solvable for $x\in T$. If, in addition, one has that $T$ is a transversal for every conjugate of $H$, then it follows that the equation $y\circ a = b$ is also uniquely solvable for $y\in T$. Thus $(T,\circ)$ is a loop. Conversely, it is not hard to show that every loop arises in this way.
The tables you are talking about remind me of this, except that you are keeping track of double coset clusters and you are not worrying about which products of representatives themselves have representatives. It would be interesting to know how this idea relates to loop theory.
