Is a group uniquely determined by the sets {ab,ba} for each pair of elements a and b?

So far I can tell we get the following list of things. When I say we "know a subgroup," I mean we know its underlying set as a subset of $G$, not its algebraic structure. For $X,Y\subseteq G$ we can extend the definition of $F$ so that $F(X,Y)=\bigcup_{x\in X,y\in Y}F(x,y)$.

• Identity: the element $e\in G$ is the unique $x\in G$ such that $F(x,x)=\{x\}$.
• Inverses: given $g\in G$, its inverse $g^{-1}\in G$ is the unique $x\in G$ such that $F(x,g)=\{e\}$.
• Powers of elements: since we have inverses and $\{g^n\}=F(g,g^{n-1})$, we can recursively compute any integer power of an element.
• Orders of elements cyclic submonoids, cycle graph: via powers of elements.
• Torsion and $p$-torsion: via orders of elements.
• Conjugacy classes: since $F(b,F(a,b^{-1}))=\{bab^{-1},a,b^{-1}ab\}$, the conjugacy class of a given element $a\in G$ is given by $\bigcup_{b\in G}F(b,F(a,b^{-1}))$.
• Commuting pairs of elements: $a,b\in G$ commute iff $F(a,b)$ is a singleton.
• Centralizers of subsets and center: follows from commuting pairs of elements.
• Cyclic orbits under inner automorphisms: given $b\in G$, the orbit of $G$ under the map $x\mapsto bxb^{-1}$ is given by applying the formula in conjugacy classes recursively.
• Normalizers of subsets and normal subsets: given a subset $X\subseteq G$, its normalizer is the set of all elements $g\in G$ for which $gXg^{-1}=X$. This is equivalent to $X$ being a union of orbits under the inner automorphism $x\mapsto gxg^{-1}$, which we know. Normal subsets follow.
• Subgroups generated by subsets: Given a subset $S$, the subgroup $\langle S\rangle$ is the intersection of all subsets of $G$ containing $S$, closed under inverses, and such that $x,y\in S\Rightarrow F(x,y)\subseteq S$.
• Lattice of subgroups and normal subgroups: we can identify which subsets are subgroups using subgroup generated by property ($S$ is a subgroup iff $S=\langle S\rangle$), then order them according to inclusion. By testing for the subgroup property and normal subset property both, we can identify normal subgroups too.
• Sylow subgroups: subgroups maximal with respect to containing $p$-torsion elements.
• Maximal/minimal (normal) subgroups: these can be identified by looking at the lattice of subgroups or lattice of normal subgroups.
• Relative normality. If $H\le K\le G$ then $H\triangleleft K$ can be tested by restricting $F_G$ to $K$ to get the function $F_K$, and then we can identify it as a normal subgroup.
• Nilpotence: nilpotent iff all maximal subgroups are normal.
• Frattini subgroup: intersect all maximal subgroups.
• Chief factors and chief series: use lattice of normal subgroups.
• Fitting subgroup: to compute the centralizer of a chief factor $H/K$, first partition $H$ into cosets of $K$ via $aK=F_H(a,K)$ (since $K$ is normal in $H$), then find all $g\in G$ such that each coset of $K$ is stable under conjugation by $g$, i.e. is a union of cyclic orbits. Then intersect all centralizers of chief factors for the fitting subgroup.
• Socle: generate a subgroup from the union of minimal normal subgroups.
• Set of commutators. Commutators are of the form $(ab)(ba)^{-1}$. So write $F(a,b)=\{x,y\}$ and then compute $H(a,b)=F(x,y^{-1})\cup F(x^{-1},y)$. The set of commutators is $\bigcup_{a,b\in G}H(a,b)$.
• Derived subgroup and derived series: used subgroup generated by and set of commutators to get $G'$. Since $F$ restricted to $G'$ is $F_{G'}$, simply iterated to get the derived series.
• Solvability: follows from derived series.

It's been suggested that we take $G$ a nonabelian group and $H,K$ two nonisomorphic groups of the same order and consider $G\times H$ versus $G\times K$. In this case, since the cycle data of $H$ and $K$ will be different, the cycle data of $G\times H$ and $G\times K$ will differ, so their $F$'s will be different if $G,H,K$ are finite, and so this does not create any finite counterexamples.