diff --git a/lib/grppcaut.gi b/lib/grppcaut.gi
index df8f7140e9..76595bf4ee 100644
--- a/lib/grppcaut.gi
+++ b/lib/grppcaut.gi
@@ -1228,14 +1228,42 @@ local spaceincl,outvecs,l,sub,yet,i,j,k,s,t,new,incl,min,rans,sofar,done,
   return a;
 end;
 
-
+PcgsCharacteristicTails:=function(G,aut)
+local gens,ser,new,pcgs,f,mo,i,j,k,s;
+  gens:=GeneratorsOfGroup(aut);
+  ser:=InvariantElementaryAbelianSeries(G,gens);
+  new:=[G];
+  for i in [2..Length(ser)] do
+    pcgs:=ModuloPcgs(ser[i-1],ser[i]);
+    f:=GF(RelativeOrders(pcgs)[1]);
+    mo:=List(gens,x->List(pcgs,y->ExponentsOfPcElement(pcgs,y^x))*One(f));
+    mo:=GModuleByMats(mo,f);
+    for j in
+      Reversed(Filtered(MTX.BasesCompositionSeries(mo),
+        x->Length(x)<Length(pcgs))) do
+      s:=ser[i];
+      for k in j do
+	s:=ClosureSubgroupNC(s,PcElementByExponents(pcgs,List(k,Int)));
+      od;
+      Add(new,s);
+    od;
+  od;
+  # build pcgs
+  ser:=[];
+  for i in [2..Length(new)] do
+    pcgs:=ModuloPcgs(new[i-1],new[i]);
+    Append(ser,pcgs);
+  od;
+  pcgs:=PcgsByPcSequence(FamilyObj(One(G)),ser);
+  return pcgs;
+end;
 
 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,
-	  quotimg,eN,field,act,spaces,sporb;
+	  quotimg,eN,field,act,spaces,sporb,npcgs,nM;
 
     asAutom:=function(sub,hom) return Image(hom,sub);end;
 
@@ -1510,7 +1538,21 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
             Info( InfoAutGrp, 2,"compute compatible pairs in group of size ",
                                   Size(A), " x ",Size(B),", ",
 				  Length(GeneratorsOfGroup(D))," generators");
-            C := CompatiblePairs( F, M, D );
+            
+            if Size(D)>10^10 then
+	      # translate to different pcgs to make tails A-invariant
+	      npcgs:=PcgsCharacteristicTails(F,A);
+	      C:=GroupWithGenerators(npcgs);
+	      SetPcgs(C,npcgs);
+	      Assert(1,Pcgs(C)=npcgs); # ensure no magic took place
+	      as:=GroupHomomorphismByImagesNC(F,Group(M.generators),
+	            Pcgs(F),M.generators);
+              nM:=rec(field:=M.field,dimension:=M.dimension,
+		      generators:=List(npcgs,x->ImagesRepresentative(as,x)));
+	      C:=CompatiblePairs(C,nM,D);
+	    else
+	      C := CompatiblePairs( F, M, D );
+	    fi;
         else
             #Info( InfoAutGrp, 2,"compute reduced gl ");
             #B := MormalizingReducedGL( spec, s, n, M );
diff --git a/lib/grppcext.gi b/lib/grppcext.gi
index e39e918183..c20ced0a96 100644
--- a/lib/grppcext.gi
+++ b/lib/grppcext.gi
@@ -831,6 +831,7 @@ local G, M, Mgrp, oper, A, B, D, translate, gens, genimgs, triso, K, K1,
 	u:=SubgroupNC(G,pcgs{idx});
       until ForAll(GeneratorsOfGroup(u),
 	i->ForAll(GeneratorsOfGroup(A),j->Image(j,i) in u));
+
       Ggens:=InducedPcgsByPcSequence(pcgs,pcgs{idx});
       tup:=M.generators{idx};
 
@@ -866,7 +867,7 @@ local G, M, Mgrp, oper, A, B, D, translate, gens, genimgs, triso, K, K1,
 	# not being perfect is not likely to do a hash conflict
 	pows:=List([0..Length(tup)],x->pows^x);
 
-	tmp:=[D, rec(hashfun:= lst->lst*pows),tup, gens,genimgs, f ];
+	tmp:=[D, rec(hashfun:= lst->lst*pows),tup, gens,newimgs, f ];
 
 	#  use `op' to get in the fake domain with the hashfun
 	tmp := StabilizerOp( D, rec(hashfun:= lst->lst*pows),tup,