Let $G$ be a finite group and $H$ a core-free subgroup. Consider the coset set $G/H$.
Assume that we have $G/H$ on which we can do whatever we want, but no more $G$ (and $H$) which is forgotten or hidden. Can we recover $G$ (and eventually $H$)? How?
By "we can do whatever we want" I mean everything (properties, operations) for which the cosets are considered as elements of $G/H$ but not as subset of $G$ (otherwise we obviously just need to take the union of all the cosets to recover $G$). For example, we can consider the multiplication of two such elements as a set of elements (exactly as the multiplication of two cosets $gH g'H$ decomposes as a disjoint union of cosets).
Note that the core-free assumption is necessary to avoid obvious counter-examples.
I checked by computer that if $G=A_4$ with $H=A_3$, then the product of two cosets $(gH)(g'H)$ is the single coset $gH$ when $g'H=H$, and otherwise it is the union of the three cosets that are not equal to $gH$.
The same is true when $G=S_4$ and $H=S_3$ so, if I have understood the question correctly, then the answer is no, it is not possible to recover $H$ and $G$ from this information.
After thinking about it some more (rather than doing computations), any two doubly transitive groups of the same degree with behave in the same way as this example, and so will also provide counterexamples. So, for example, there are two such groups with$|G| = 72$ and $|H|=8$.