Normal subgroups of p-groups

Question

Let $G$ be a group of order $p^\alpha$, where $p$ is prime. If $H\lhd G$, then can we find a normal subgroup of $G/H$ that has order $p$?

Answer 1

Theorem: a group $\,G\,$ of order $\,p^n\,,\,p\,$ a prime, $\,n\in\mathbb N\,$ , always has normal subgroup of order $\,p^m\,\,,\,\,\forall\, m\leq n\,\,,\,m\in \mathbb N$

Proof: Exercise, using that always $|Z(G)|>1\,$ and induction on $\,n\,$

So the comments by Gerry and Geoff close the matter.