Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
The dihedral group of order $8$ - the symmetries of a square is one such.
It has the identity element and two elements of order $4$ (clockwise and counterclockwise quarter turns). The other five elements are four reflections about axes of symmetry and a half turn - all of order 2.
Consider $D_4$ Dihedral group, non-abelian, $5$ elements of order $2$.