How to determine regular subgroup

Question

I understand this has to do with the structure of p-groups.

• Let $G$ be a group acting transitively on a set $X$ where $|X|=p^{n+1}$.

• Suppose that there is a subgroup $H$ of $G$ acting transitively on $X$ and that $|H|=p^{n+2}$. The stabiliser $H_x$ of a point $x$ in $H$ is such that it is not normal in $H$ and that $|H_x|=p$.

I don't know how this information can enable me arrive at a conclusion to the following question:

Prove that $G$ has no subgroup that acts regularly on $X$.

Answer 1

Counterexample: take $G=H=D_4$, the dihedral group of order $8$ acting naturally on $4$ points. (So $p=2$ and $n=1$.) The stabiliser has order $2$ and is not normal, but there are regular subgroups.