Conjugacy-closed not implies weak subset-conjugacy-closed

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., conjugacy-closed subgroup) need not satisfy the second subgroup property (i.e., weak subset-conjugacy-closed subgroup)
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Statement

It is possible to have a group $G$ and a subgroup $H$ of $G$ such that $H$ is a conjugacy-closed subgroup of $G$ but not a weak subset-conjugacy-closed subgroup of $G$. In particular, $H$ is not a Subset-conjugacy-closed subgroup (?) of $G$.

Facts used

1. Brauer's permutation lemma
2. Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

Proof

Consider $G=GL(6,\mathbb {Q} )$ and $H$ as the symmetric group of degree six, embedded in $G$ via permutation matrices. By fact (1), $H$ is conjugacy-closed in $G$, and by fact (2), $H$ is not subset-conjugacy-closed in $G$.

Analogous examples can be constructed by replacing $\mathbb {Q}$ with a finite field whose characteristic is a large enough prime.