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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.

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The name you are looking for is that of a torsor or principal homogeneous space. For any sets $X$ and $Y$, the set of bijections $X \to Y$ is a torsor for the group $\mathrm{Sym}(X)$ acting on the right as well as the group $\mathrm{Sym}(Y)$ acting on the left. In your case, the set $\Phi$ is a torsor for the abelian group you constructed.

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What you are looking at is called a heap or groud. The category of heaps is equivalent to the categor of torsors but is the universal algebra viewpoint from the sixties. The wiki article I linked to gives your exact construction of a heap as a main example. The nice thing about the heap viewpoint is you don't need to know explicitly the associated group.

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    $\begingroup$ Kontsevich uses this definition in "Operads and Motives in Deformation Quantization". $\endgroup$
    – Spice the Bird
    Commented Jan 10, 2012 at 21:46

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