Thanks for any help or comments.


Suppose that $G$ is a finite group. A *Carter subgroup* of $G$ is a nilpotent
self-normalizing subgroup of $G$. Carter and Vdovin have shown that solvable
groups have Carter subgroups, and that in addition, in every group with
Carter subgroups, the Carter subgroups are conjugate -- see 


Carter, R. W. (1961), [*Nilpotent selfnormalizing subgroups of soluble
groups*](https://dx.doi.org/10.1007%2FBF01211016), Mathematische Zeitschrift, 
75 (2): 136–139.


Vdovin, E. P. (2006), [*On the conjugacy problem for Carter subgroups.
(Russian.)*](https://www.ams.org/mathscinet-getitem?mr=2265277), Sibirsk. 
Mat. Zh., 47 (4): 725–730. Translation in Siberian
Math. J. 47 (2006), no. 4, 597–600


Vdovin, E. P. (2007), [*Carter subgroups in finite almost simple groups. (Russian.)*](https://www.ams.org/mathscinet-getitem?mr=2356523), Algebra i Logika, 46 (2): 157–216.


My question is about the structure of groups whose Carter subgroups are
their Sylow $2$-subgroups? I mean, I am interested in any theorem which
guides me towards some classification of this type of groups.



Finite groups whose Carter subgroups are the Sylow 2-Subgroups
Classifying finite groups with self-normalizing Sylow $2$-subgroup seems rather
hopeless as $10608361$ of the $10625619$ groups of order less than $768$ have
this property. --


A [GAP](http://www.gap-system.org) function to count the groups of order $n$ with this property is
as follows:



```
NrOfGroupsWithSelfNormalizingSylow2Subgroup := function ( n )

  if   n = 1 then return 1;
  elif SmallestRootInt(n) = 2 then
    return NrSmallGroups(n);
  elif n mod 2 = 1 then
    return 0;
  else
    return Number(AllGroups(n),
                  G -> Size(SylowSubgroup(G,2))
                     = Size(Normalizer(G,SylowSubgroup(G,2)))); 
  fi;
end;

```

