Converse of congruence condition on Sylow numbers for the prime two

Statement
For any odd number $$m$$, there exists a finite group $$G$$ where the number of 2-fact about::Sylow subgroups of $$G$$ (i.e., the fact about::Sylow number for the prime $$2$$) equals $$m$$.

Proof
We take $$G$$ as the dihedral group $$D_{2m}$$ of degree $$m$$ and order $$2m$$. It is given by the presentation:

$$G := \langle a,x \mid a^m = x^2 = e, xax = a^{-1} \rangle$$.

The two-element subgroup $$\langle x \rangle$$ is a 2-Sylow subgroup, and it is a self-normalizing subgroup. It has $$m$$ conjugates, given by $$\langle a^r x \rangle$$, with $$0 \le r \le m - 1$$.