Paranormal not implies pronormal

Statement
A paranormal subgroup of a group need not be pronormal.

Pronormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed pronormal in $$G$$ if for any $$g \in G$$ there exists $$x \in \langle H, H^g \rangle$$ such that $$H^x = H^g$$.

Paranormal subgroup
A subgroup $$H$$ of a group $$G$$ is termed paranormal in $$G$$ if for any $$g \in G$$, $$H$$ is a contranormal subgroup of $$\langle H, H^g \rangle$$: in other words, the normal closure of $$H$$ in $$\langle H, H^g \rangle$$ is the whole group $$\langle H, H^g$$.

Example of the symmetric group on six elements
Let $$G$$ be the symmetric group on the set $$\{ 1,2,3,4,5,6 \}$$. Then there are in fact four different conjugacy classes of subgroups that are paranormal but not pronormal. We list these examples (by providing a representative subgroup for each) and explain why each one works:


 * 1) $$H_1 := \langle (4,5,6), (2,3)(5,6) \rangle$$:
 * 2) * This is paranormal
 * 3) * This is not pronormal
 * 4) $$H_2 := \langle (1,2,3)(4,5,6), (2,3)(5,6) \rangle$$:
 * 5) * This is paranormal
 * 6) * This is not pronormal
 * 7) $$H_3 := \langle (1,2,3), (2,3), (4,5) \rangle$$. In other words, $$H_3$$ is the product of the symmetric group on the first three elements and the symmetric group on the fourth and fifth elements.
 * 8) * This is paranormal
 * 9) * This is not pronormal
 * 10) $$H_4 := \langle (1,2)(3,4)(5,6), (1,3,5)(2,4,6), (3,5)(4,6) \rangle$$
 * 11) * This is paranormal
 * 12) * This is not pronormal