Let $\Phi$ be a set of bijections $\phi_a:X\to Y$. To each pair of bijections $\phi_a$, $\phi_b$ one naturally relates a bijection $\psi_{ab}:=\phi_a^{-1}\circ\phi_b: X\to X$. In some cases the set of all such $\psi_{ab}$ forms a subgroup of $Sym(X)$, the group of all bijections $X\to X$.

Were these kinds of constructions studied in any generality? The question arose as in our very special setting, with $X$ finite, the resulting group turned out to be abelian. At least it would be nice to know if this kind of construction has a name.