Conjugacy-closed not implies weak subset-conjugacy-closed

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 fact about::subset-conjugacy-closed subgroup of $$G$$.

Facts used

 * 1) uses::Brauer's permutation lemma
 * 2) uses::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.