Finite groups whose Carter subgroups are the Sylow 2-Subgroups 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, Mathematische Zeitschrift, 
75 (2): 136–139.
Vdovin, E. P. (2006), On the conjugacy problem for Carter subgroups.
(Russian.), 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.), 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.
 A: Here's some stronger evidence in favor of Stefan's claim that such groups are too numerous and varied to expect a classification.
glist := AllSmallGroups(Size,[2..511],IsPGroup,false,G->Size(G) mod 2 = 0, true,
           G->SylowSubgroup(G,2)=Normalizer(G,SylowSubgroup(G,2)));;
Length(glist);
Sum(List(Filtered([2..511],n->(n mod 2 = 0) and 
    (not IsPrimePowerInt(n))),n->NrSmallGroups(n)));

This shows that of the 33510 non-p-groups with even order less than 512, 25673 of them have self-normalizing Sylow 2-subgroups.  Moreover, given $n\leq 7$, every 2-group of order $2^n$ appears as the Sylow 2-subgroup in some such example.
syls:=Set(List(glist,G->IdGroup(SylowSubgroup(G,2))));;
ForAll([1..7],n->NrSmallGroups(2^n) = Number(syls,P->P[1]=2^n));

So not only does it look like we can expect a large proportion of groups to have this property, but that we cannot even reasonably expect to constrain the structure of the Sylow 2-subgroup.
A: 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 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;

