Iterating over an automorphism group

How can we iterate over the elements of an automorphism group?

Let us assume we need to iterate over the six automorphisms of the Klein group.

> G := AbelianGroup([2,2]);
> A := AutomorphismGroup(G);

The naive idea doesn’t work. Typing

> { a : a in A };

produces an error message, saying: Iteration is not possible over this object. Thus one cannot directly iterate over automorphism groups. We first need to convert the automorphism group into a structure suitable for iteration. PermutationGroup and FPGroup do not work, and something strange happens with PCGroup. So what we should do? Here is the trick:

> p, P := PermutationRepresentation(A);

Here, p is a (bijective) map from A onto the permutation group P. We can iterate over P, and we need to use the inverse of the map p to identity our elements inside A.

> { x @@ p : x in P };
{
    Automorphism of Abelian Group isomorphic to Z/2 + Z/2
    Defined on 2 generators
    Relations:
        2*G.1 = 0
        2*G.2 = 0 of order 1 which maps:
        G.1 |--> G.1
        G.2 |--> G.2,
    Automorphism of Abelian Group isomorphic to Z/2 + Z/2
    Defined on 2 generators
    Relations:
        2*G.1 = 0
        2*G.2 = 0 which maps:
        G.1 |--> G.1
        G.2 |--> G.1 + G.2,
    Automorphism of Abelian Group isomorphic to Z/2 + Z/2
    Defined on 2 generators
    Relations:
        2*G.1 = 0
        2*G.2 = 0 which maps:
        G.1 |--> G.2
        G.2 |--> G.1,
    Automorphism of Abelian Group isomorphic to Z/2 + Z/2
    Defined on 2 generators
    Relations:
        2*G.1 = 0
        2*G.2 = 0 which maps:
        G.1 |--> G.1 + G.2
        G.2 |--> G.2,
    Automorphism of Abelian Group isomorphic to Z/2 + Z/2
    Defined on 2 generators
    Relations:
        2*G.1 = 0
        2*G.2 = 0 which maps:
        G.1 |--> G.2
        G.2 |--> G.1 + G.2,
    Automorphism of Abelian Group isomorphic to Z/2 + Z/2
    Defined on 2 generators
    Relations:
        2*G.1 = 0
        2*G.2 = 0 which maps:
        G.1 |--> G.1 + G.2
        G.2 |--> G.1
}