How to list automorphisms on a finite set and analyze them

Question

Let s be a finite set of integer lattice points, for example s = { {0,0} , {0,3} , {1,1} , {0,2} }.

I am trying to do two things:

1. Create a list of all the automorphisms on s. That is, if $s$ is our set, I want to list all onto functions $f:s \rightarrow s$, not including duplicates. (Consider $f$ and $g$ duplicates if for each point in $s$, that point is taken by $f$ and $g$ to the same point.)
2. Among the automorphisms, find those $f$ for which Condition[ x , f[x] ] is True for all x in s.

Here, Condition is just an aribtrary function that takes in two $2$-tuples and returns True or False.

I am trying to do (1) to accomplish (2). But, I have never tried to handle lists of functions before, and am not sure how to do it in Mathematica. I am trying to do this by treating a 'function' as a collection of $2$-tuples of $2$-tuples, where the second element is the target of the function acting on the first. But this is very, very, tedious, and I am hoping there is a better way.

Answer 1

To make it simple, I choose a set of symbols (you can replace them with points in 1,2,3.. n dimensions):

sym = Array[Subscript[s, #] &, 4]

We get all automorphism by Permutations[sym]. A permutation gives the image of the first element, the second element ...

automorph= Permutations[sym]

Finally to pick all automorphism that fulfill some predicate "pred" we can use "Select". E.g. to get all automorphism that leave the first 2 elements in place we define the predicate:

pred[x_] := x[[1]] === sym[[1]] && x[[2]] === sym[[2]]

Or all together:

sym = Array[Subscript[s, #] &, 4];
automorph = Permutations[sym];
pred[x_] := x[[1]] === sym[[1]] && x[[2]] === sym[[2]]
Select[automorph, pred]