Dear Masoud,
You could look the following GAP session:
gap> G:=DihedralGroup(32);
gap> H:=NormalSubgroups(G)[2]; ## H is a normal subgroup of G
gap> p:=NaturalHomomorphismByNormalSubgroup(G,H); ## p: G->G/H
gap> GH:=Range(p); ## GH is the quotient G/H
gap> A:=AutomorphismGroup(G); ## A=AutG
gap> a:=Random(A); ## f is a element in AutG
gap> gH:=List(GH,x->PreImagesRepresentative(p,x)); ##gH is the list of
representatives {g_1,g_2,...,g_n}
gap> agH:=List(gH,x->x^a); ## agH:={g_1^a,g_2^a,...,g_n^a}
gap> aGH:=List(agH,x->x^p);
Then aGH is the list that you want to compute
Best regards,
Luyen
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