I am interested in the following problem: given a finite group $G$, and elements $a, b, c, d \in G$, what are the functions $f:G \to G$ such that $f(a) \cdot f(b) = f(c) \cdot f(d)$ if $a \cdot b = c \cdot d$? Here, $\cdot$ is the product in the group, and note that I do not require the identity element to be preserved under the function $f$. Essentially, I am asking how much freedom I have to relabel the elements of the group without changing the product of two elements.
I have worked through all of the finite groups of order 5 or less, but I was wondering if there were general results beyond that point, such as when the group in question is non-abelian. The maps $f$ should be in the symmetric group acting on $|G|$ elements, but how does the subgroup of $S_{|G|}$ depend on the choice of $G$?