diff --git a/gap/perm.gi b/gap/perm.gi
index 376ac1b05..a147835d7 100644
--- a/gap/perm.gi
+++ b/gap/perm.gi
@@ -350,7 +350,9 @@ end;
 #! @BeginChunk ThrowAwayFixedPoints
 #! This method defines a homomorphism of a permutation group
 #! <A>G</A> to the action on the moved points of <A>G</A> if
-#! <A>G</A> does not have too many moved points. In the current setup, the
+#! either <A>G</A> does not have too many moved points, or if <A>G</A> is
+#! primitive and has fixed points.
+#! In the current setup, the
 #! homomorphism is defined if the number <M>k</M> of moved
 #! points is at most <M>1/3</M> of the largest moved point of <A>G</A>,
 #! or  <M>k</M> is at most half of the number of points on which
@@ -361,12 +363,15 @@ end;
 FindHomMethodsPerm.ThrowAwayFixedPoints :=
   function( ri, G )
       # Check, whether we can throw away fixed points
-      local gens,hom,l,n,o;
+      local gens,hom,l,n,largest,o;
 
       gens := GeneratorsOfGroup(G);
       l := List(gens,StoredPointsPerm);
       n := NrMovedPoints(G);
-      if 2*n > Maximum(l) or 3*n > LargestMovedPoint(G) then  # we do nothing
+      largest := LargestMovedPoint(G);
+      if (2*n > Maximum(l) or 3*n > largest)
+              and not (IsPrimitive(G) and largest > n)
+      then  # we do nothing
           return NeverApplicable;
       fi;
       o := MovedPoints(G);