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Let $G$ be a finite group and $H$ be a subgroup of $G$. For any prime power order element $x$ of $G$, if there exsits some element $g$ in $G$ such that $x^g$ is contained in $H$, can we get $G=H$?

Let $G$ be a finite group and $H$ be a subgroup of $G$. Suppose that for any prime power order element $x$ of $G$, there exists some element $g$ in $G$ such that $x^g$ is contained in $H$. Does it follow that $H=G$?