diff --git a/lib/autsr.gi b/lib/autsr.gi
index f233ac004f..0a09683580 100644
--- a/lib/autsr.gi
+++ b/lib/autsr.gi
@@ -1690,7 +1690,7 @@ end);
 # isomorphism available and there are many generators
 InstallGlobalFunction(PatheticIsomorphism,function(G,H)
 local d,a,map,cG,nG,nH,i,j,u,v,asAutomorphism,K,L,conj,e1,e2,
-      iso,api,gens,pre,aab,as;
+      iso,api,gens,pre,aab,as,somechar;
 
   asAutomorphism:=function(sub,hom)
     return Image(hom,sub);
@@ -1784,11 +1784,16 @@ local d,a,map,cG,nG,nH,i,j,u,v,asAutomorphism,K,L,conj,e1,e2,
   aab:=[Image(e1,G),Image(e2,H)];
   # we also fix the *pairs* of the characteristic subgroups as orbits. Again
   # this must happen in Aut(G)\wr 2, and reduces the size of the group.
-  a:=AutomorphismGroup(d:autactbase:=aab,someCharacteristics:=
-    rec(subgroups:=cG,
-        orbits:=List([1..Length(nG)],x->[Image(e1,nG[x]),Image(e2,nH[x])])),
+  somechar:=rec(subgroups:=cG,
+        orbits:=List([1..Length(nG)],x->[Image(e1,nG[x]),Image(e2,nH[x])]));
+  a:=AutomorphismGroup(d:autactbase:=aab,someCharacteristics:=somechar,
     directs:=aab,
     delaypermrep:=true );
+  for i in cG do
+    if not ForAll(GeneratorsOfGroup(a),x->Image(x,i)=i) then
+      a:=Stabilizer(a,i,asAutomorphism);
+    fi;
+  od;
 
   iso:=fail;
   #if NrMovedPoints(api)>5000 then
@@ -1815,6 +1820,7 @@ local d,a,map,cG,nG,nH,i,j,u,v,asAutomorphism,K,L,conj,e1,e2,
     as:=a;
     Add(cG,TrivialSubgroup(d));
 
+    SortBy(cG,x->-Size(x));
     for i in cG do
       u:=ClosureGroup(i,K);
       v:=ClosureGroup(i,L);
@@ -1857,10 +1863,12 @@ local d,a,map,cG,nG,nH,i,j,u,v,asAutomorphism,K,L,conj,e1,e2,
           conj:=conj*map;
           K:=Image(map,K);
 
-          u:=Stabilizer(api,v,gens,pre,asAutomorphism);
-          Info(InfoMorph,1,"Factor ",Size(d)/Size(i),": ",
-              "reduce by ",Size(api)/Size(u));
-          api:=u;
+          if Size(i)>1 then
+            u:=Stabilizer(api,v,gens,pre,asAutomorphism);
+            Info(InfoMorph,1,"Factor ",Size(d)/Size(i),": ",
+                "reduce by ",Size(api)/Size(u));
+            api:=u;
+          fi;
         fi;
       fi;
     od;
diff --git a/lib/morpheus.gi b/lib/morpheus.gi
index 67f7d5ad8f..dd241b6a05 100644
--- a/lib/morpheus.gi
+++ b/lib/morpheus.gi
@@ -2860,7 +2860,8 @@ local m;
   fi;
 
   if Size(SolvableRadical(G))>1 and CanComputeFittingFree(G)
-    and not (IsSolvableGroup(G) and Size(G)<=2000)
+    and not (IsSolvableGroup(G) and Size(G)<=2000
+       and ID_AVAILABLE(Size(G))<>fail)
     and (AbelianRank(G)>2 or Length(SmallGeneratingSet(G))>2
       # the solvable radical method got better, so force if the radical of
       # the group is a good part
diff --git a/tst/testextra/grpauto.tst b/tst/testextra/grpauto.tst
index 77d60ef8bd..1c878b29fd 100644
--- a/tst/testextra/grpauto.tst
+++ b/tst/testextra/grpauto.tst
@@ -12,7 +12,7 @@ gap> SetAssertionLevel(0);
 #
 gap> G:=PcGroupCode(2664057732308087629446918084678644731357911420184737672,
 > 6272);;
-gap> H:=PcGroupCode(849169324138369034371257796152864931485540637536648, 
+gap> H:=PcGroupCode(849169324138369034371257796152864931485540637536648,
 > 6272);;
 gap> IsomorphismGroups(G,H);
 fail
@@ -88,7 +88,7 @@ gap> G:=Group((1,3,8,21,37,43,36,35)(2,6,15,30,41,29,19,4)(5,7,18,34,46,
 > (56,57),(1,2,5,12,26)(3,7,17,6,14)(4,10,25,35,42)(8,20,9,18,15)
 > (11,27,38,19,36)(13,29,28,43,32)(21,23,30,24,34)(33,37,41,46,39)
 > (49,50,52,51,53));;
-gap> Size(AutomorphismGroup(G)); 
+gap> Size(AutomorphismGroup(G));
 2880000
 
 #
@@ -105,7 +105,7 @@ false
 
 # Isomorphism tests
 gap> gp1:=Group(
-> (1,23,6,64,38)(2,42,18,19,11)(3,7,30,49,50)(4,14,20,45,46)(5,9,21,41,58,34, 
+> (1,23,6,64,38)(2,42,18,19,11)(3,7,30,49,50)(4,14,20,45,46)(5,9,21,41,58,34,
 > 17,29,48,44)(8,26,40,43,33)(10,22,59,60,32)(12,24,53,74,76)(13,31,55,81,77,
 > 61,25,52,75,83)(15,27,51,78,79)(16,28,47,68,71,62,36,56,70,69)(35,57,82,92,
 > 91,37,54,80,90,93)(66,72)(85,86)(88,94)(95,96)(97,98,100,102,104)(99,101,
@@ -113,14 +113,14 @@ gap> gp1:=Group(
 > 27,14,24)(9,25,17,31)(16,35,62,37)(18,63,64,39)(20,46,26,43)(21,44,29,
 > 58)(22,60,30,50)(28,54,36,57)(40,67,45,65)(41,66,48,72)(47,69,56,71)(49,84,
 > 59,73)(51,79,53,76)(52,77,55,83)(68,85,70,86)(74,89,78,87)(75,88,81,94)(80,
-> 91,82,93)(90,95,92,96)(97,98)(99,101)(102,104)(105,106), 
+> 91,82,93)(90,95,92,96)(97,98)(99,101)(102,104)(105,106),
 > (4,33)(8,14)(12,15)(16,62)(20,26)(24,27)(28,36)(35,37)(40,45)(43,46)(47,
 > 56)(51,53)(54,57)(65,67)(68,70)(69,71)(74,78)(76,79)(80,82)(85,86)(87,
 > 89)(90,92)(91,93)(95,96), (1,4)(2,8)(3,12)(5,16)(6,20)(7,24)(9,28)(10,
 > 27)(11,33)(13,35)(14,23)(15,32)(17,36)(18,40)(19,43)(21,47)(22,51)(25,
 > 54)(26,42)(29,56)(30,53)(31,57)(34,62)(37,61)(38,46)(39,65)(41,68)(44,
 > 69)(45,64)(48,70)(49,74)(50,76)(52,80)(55,82)(58,71)(59,78)(60,79)(63,
-> 67)(66,85)(72,86)(73,87)(75,90)(77,91)(81,92)(83,93)(84,89)(88,95)(94,96), 
+> 67)(66,85)(72,86)(73,87)(75,90)(77,91)(81,92)(83,93)(84,89)(88,95)(94,96),
 > (1,5)(2,9)(3,13)(4,16)(6,21)(7,25)(8,28)(10,31)(11,34)(12,35)(14,36)(15,
 > 37)(17,23)(18,41)(19,44)(20,47)(22,52)(24,54)(26,56)(27,57)(29,42)(30,
 > 55)(32,61)(33,62)(38,58)(39,66)(40,68)(43,69)(45,70)(46,71)(48,64)(49,
@@ -132,7 +132,7 @@ gap> gp2:=Group(
 > 83)(87,89)(88,94), (1,2,6,21,40,11,26,44,64,22)(3,10,34,61,53,36,7,25,52,
 > 62)(4,14,23,47,48)(5,9,24,43,60,20,18,32,50,46)(8,29,42,45,16)(12,27,56,74,
 > 76)(13,35,58,81,77,39,28,55,75,83)(15,30,54,78,79)(17,33,49,69,71)(19,31,
-> 51,68,70)(37,57,82,90,92)(38,59,80,91,93)(41,63)(66,72)(73,84)(88,94), 
+> 51,68,70)(37,57,82,90,92)(38,59,80,91,93)(41,63)(66,72)(73,84)(88,94),
 > (1,4,11,16)(2,8,26,14)(3,12,36,15)(5,19,20,17)(6,23,44,29)(7,27,10,30)(9,33,
 > 18,31)(13,38,39,37)(21,42,64,47)(22,45,40,48)(24,51,32,49)(25,54,34,56)(28,
 > 59,35,57)(41,65,63,67)(43,69,50,68)(46,71,60,70)(52,74,61,78)(53,76,62,
@@ -141,7 +141,7 @@ gap> gp2:=Group(
 > 33)(15,38)(16,19)(18,26)(21,43)(22,46)(23,49)(25,55)(27,57)(29,51)(30,
 > 59)(32,44)(34,58)(36,39)(40,60)(41,66)(42,68)(45,70)(47,69)(48,71)(50,
 > 64)(52,75)(53,77)(54,80)(56,82)(61,81)(62,83)(63,72)(65,85)(67,86)(73,
-> 88)(74,90)(76,92)(78,91)(79,93)(84,94)(87,95)(89,96), 
+> 88)(74,90)(76,92)(78,91)(79,93)(84,94)(87,95)(89,96),
 > (1,36,11,3)(2,10,26,7)(4,15,16,12)(5,39,20,13)(6,40,44,22)(8,30,14,27)(9,35,
 > 18,28)(17,38,19,37)(21,63,64,41)(23,48,29,45)(24,60,32,46)(25,62,34,53)(31,
 > 59,33,57)(42,67,47,65)(43,72,50,66)(49,71,51,70)(52,84,61,73)(54,79,56,
@@ -188,4 +188,10 @@ gap> gp2:=Group([(1,6,3)(2,5,4)(7,23,25)(8,24,26)(9,19,31)
 > 83,82,77)(73,99,91,81,89,79,93,98,95,78)(80,92)]);;
 gap> IsomorphismGroups(gp1,gp2)<>fail;
 true
+
+# MEssy p-groups, forum by D.Robinson, 4/5/23
+gap> g1:=Image(IsomorphismPcGroup(AutomorphismGroup(SmallGroup(64,112))));;
+gap> g2:=SmallGroup(512,10493360);;
+gap> IsomorphismGroups(g1,g2)<>fail;
+true
 gap> STOP_TEST( "grpauto.tst", 1);