ENHANCE: Use autactbase to create better permrep of factor automorphi…

…sms.

lib/grppcaut.gi

@@ -1262,7 +1262,7 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
     local spec, weights, first, m, pcgsU, F, pcgsF, A, i, s, n, p, H, 
           pcgsH, pcgsN, N, epi, mats, M, autos, ocr, elms, e, list, imgs,
           auto, tmp, hom, gens, P, C, B, D, pcsA, rels, iso, xset,
-          gensA, new,as,somechar,scharorb,asAutom,autactbase,
+          gensA, new,as,somechar,scharorb,asAutom,actbase,
 	  quotimg,eN,field,act,spaces,sporb,npcgs,nM;
 
     asAutom:=function(sub,hom) return Image(hom,sub);end;
@@ -1273,8 +1273,8 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
         x->PcElementByExponents(pcgs,ExponentsOfPcElement(spec,x){[1..Length(pcgs)]})));
     end;
 
-    autactbase:=ValueOption("autactbase");
-    PushOptions(rec(autactbase:=fail)); # remove this option from concern
+    actbase:=ValueOption("autactbase");
+    PushOptions(rec(actbase:=fail)); # remove this option from concern
     somechar:=ValueOption("someCharacteristics");
     if somechar<>fail then
       scharorb:=somechar.orbits;
@@ -1390,9 +1390,23 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
     A     := AutomorphismGroupElAbGroup( F, B );
     SetIsGroupOfAutomorphismsFiniteGroup(A,true);
 
+    # for first step
+    H:=F;
+    pcgsH:=Pcgs(H);
+
     # run down series
     for i in [2..Length(first)-1] do
 
+	if Length(GeneratorsOfGroup(A))>0 and not HasNiceMonomorphism(A) then
+	  if Source(A.1)<>H then
+	    Error("shit");
+	  fi;
+	  if actbase<>fail then
+	    e:=List(actbase,x->quotimg(H,pcgsH,x));
+	    NiceMonomorphism(A:autactbase:=e);
+	  fi;
+
+	fi;
         # get factor
         s := first[i];
         n := first[i+1];
@@ -1650,8 +1664,8 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
 	  SortBy(B,Size);
 	  SetIsGroupOfAutomorphismsFiniteGroup(A,true);
 	  tmp:=Size(A);
-	  if autactbase<>fail then
-	    e:=List(autactbase,x->quotimg(H,pcgsH,x));
+	  if actbase<>fail then
+	    e:=List(actbase,x->quotimg(H,pcgsH,x));
 	    NiceMonomorphism(A:autactbase:=e);
 	  fi;
 	  for e in B do