Pronormal not implies NE

Statement
A pronormal subgroup of a group need not be a NE-subgroup.

Facts used

 * 1) uses::Sylow implies pronormal
 * 2) uses::Sylow not implies NE

Examples of Sylow subgroups
The proof that pronormal subgroups need not be NE follows from facts (1) and (2). Further, any example of a Sylow subgroup that is not NE gives an example of a pronormal subgroup that is not NE. Two examples of situations where Sylow subgroups are not NE are the $$2$$-Sylow subgroup and the $$5$$-Sylow subgroup in the alternating group of degree five.

Example of the symmetric group
Let $$G$$ be the symmetric group on the set $$\{ 1,2,3,4 \}$$, and $$H$$ be the four-element subgroup $$\{, (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}$$. Then, $$H$$ is a pronormal subgroup, because the subgroup generated by $$H$$ and any other conjugate of it is the whole group. On the other hand, we have $$H^G = G$$ and $$N_G(H)$$ is a dihedral group of order eight, so $$H^G \cap N_G(H)$$ is a dihedral group of order eight, which is strictly bigger than $$H$$.