Use NormalizerViaRadical for suitable permutation groups (#5512)

doc/tut/group.xml

@@ -150,7 +150,7 @@ whose image is the factor group.
 <Example><![CDATA[
 gap> hom := NaturalHomomorphismByNormalSubgroup( norm, elab );
 [ (2,3)(6,7), (3,4)(7,8), (3,5)(4,6), (5,7)(6,8), (5,6)(7,8),
-  (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,5)(2,6)(3,7)(4,8)
+  (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7)
  ] -> [ (1,2)(5,6), (2,3)(6,7), (2,4)(3,5), (4,6)(5,7), (4,5)(6,7),
   (), (), () ]
 gap> f := Image( hom );

lib/stbcbckt.gi

@@ -2671,13 +2671,22 @@ Eh, Lh, Nh,G0;
     L := arg[ 3 ];
     issub:=fail;
   elif IsSubset( G, E )   then
+    if Size(G)=Size(E) then
+      return G;
+    fi;
     L := E;
     issub:=true;
   else
     L := TrivialSubgroup( G );
     issub:=false;;
   fi;
 
+  if issub and HasFittingFreeLiftSetup(G) and NrMovedPoints(G)>1000
+    # radical is at lest 3rd root of |G| -- avoid smallish radical
+    and Size(FittingFreeLiftSetup(G).radical)^3>Size(G) then
+    return NormalizerViaRadical(G,E);
+  fi;
+
   mpG:=MovedPoints(GeneratorsOfGroup(G));
   mpE:=MovedPoints(GeneratorsOfGroup(E));
   if IsSubset(mpG,mpE) and not IsTransitive(G,mpG) then