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.