The usual definition of a normal subgroup $H$ of group $G$ is that $x^{-1}Hx=H$ hold for all $x \in G.$ A weaker form of this is to assert only the containment $$x^{-1}Hx \subseteq H \tag{1}$$ should hold, for each $x \in G.$ If it does, then for any specific $x,$ equality in $(1)$ be deduced by applying it both as is, and with $x^{-1}$ replaced by $x$. Thus the "weaker" form is equivalent to the usual one, provided one assumes it holds for all $x \in G.$
My question concerns cases of a group $G$, a subgroup $H,$ and a choice of just one $x\in G.$ According to Hall's group theory text, in the case of infinite $H$ the containment $(1)$ need not imply equality.
So I'm looking for such examples: A group $G$ with a subgroup $H$ and some specific $x \in G$ for which the containment $(1)$ holds without equality. That is, we have $x^{-1}Hx \subset H,$ where the containment is proper.
