The following is Thompson’s normal $p$-complement theorem:
Let $G$ be a finite group and $S$ a Sylow $p$-subgroup. Suppose that $p$ is odd or $G$ is $S_4$-free. Then $G$ has normal $p$-complement if and only if $C_G(Z(S))$ and $N_G(J(S))$ has normal $p$-complement.
The theorem in the original paper (Thompson, 1964) only gives the part of $p$ odd. Recall that a group is called $H$-free if no section of $G$ is isomorphic to $H$. In many later references, “$S_4$-free” is added in the statement. See for example (Huppert, 1967) and (Glauberman, 1971).
Another famous normal $p$-complement theorem is the one attributed to Glauberman-Thompson:
Let $G$ be a finite group and $S$ a Sylow $p$-subgroup. Suppose that $p$ is odd. Then $G$ has normal $p$-complement if and only if $N_G(Z(J(S)))$ has normal $p$-complement.
I wonder if the $p=2$ part is valid for Glauberman-Thompson $p$-nilpotency criterion if the hypothesis “$S_4$-free” is added. But I could not find any $S_4$-free examples. Or can we do that with another functor like the one given by Stellmacher, that is, does the $p=2$ case follows immediately from any functor that admits a ZJ-type theorem?
Reference:
[1] Glauberman G. Global and local properties of finite groups. M.B. Powell and G. Higman, eds. Finite Simple Groups (Proc. Instructional Conf. Oxford, 1969). London: Academic Press, 1971, 1-64.
[2] Thompson J G. Normal $p$-complements for finite groups. J. Algebra, 1964, 43-46.