In GAP, how to generate a quotient automorphism group so that its elements are still maps?

Question

For simplicity, I've tried to work with automorphism and inner automorphism groups by using the operator /. Although this attempt effectively computes the correct group, which is called the outer automorphism group, its elements do not show up as maps. Here is a quick GAP code to view the situation:

gap> D := DihedralGroup(8);;
gap> autD := AutomorphismGroup(D);;
gap> innD := InnerAutomorphismsAutomorphismGroup(autD);;
gap> outD := autD / innD;;
gap> StructureDescription(outD);
"C2"
gap> IsMapping(outD.1);
false

Answer 1

I usually avoid the terminology "outer automorphism" because it is misleading: those are not automorphisms, they are equivalence classes of automorphisms only.

You could of course ask GAP for a representative of one of these equivalence classes; but then make sure what you are doing with it really is independent of which representative you picked...

gap> hom:=NaturalHomomorphismByNormalSubgroup(autD,innD);
CompositionMapping( [ (1,3,4,6), (1,4)(3,6), (2,5)(3,6) ] -> 
[ f1, <identity> of ..., <identity> of ... ],
 <action isomorphism> )
gap> PreImagesRepresentative(hom,outD.1);
[ f1, f2 ] -> [ f1*f2, f2 ]