For some work in equivariant stable homotopy, I am trying to understand the family of finite $p$-groups $P$ with derived subgroup $P'$ of order $p$. There is a 1999 J. Algebra paper by Simon Blackburn (Groups of prime power order with derived subgroup of prime order) that gives a very detailed classification, but I would like to understand these a bit more conceptually, and don't really care about uniqueness of description.

So I am wondering if my group theory friends can tell me (perhaps with a reference) if it is correct that all such groups can be constructed as follows:

(a) Start with an extra special $p$ group $\widetilde V$, so it sits in a nonsplit short exact sequence $$ C_p \rightarrow \widetilde V \rightarrow V,$$ where $V$ is an elementary abelian group of even dimension, and $C_p = \widetilde V^{\prime}$.

(b) Then pullback via a surjective map $\pi: A \rightarrow V$, where $A$ is an abelian $p$ group, yielding a nonsplit short exact sequence $$ C_p \rightarrow \widetilde A \rightarrow A,$$ with $C_p = \widetilde A^{\prime}$.

(c) [See Derek Holt's example, and ensuing comments.] Note that $Z(\widetilde A) = C_p \times \ker \pi$. Let $\alpha: C_p \rightarrow \ker \pi$ be a homomorphism, and let $C < Z(\widetilde A)$ be its graph. Now pushout via an inclusion $C \hookrightarrow C_{p^k}$, yielding a group $P$.

Then $P$ is a $p$-group of the sort I am interested in: $P' = C_p$. Furthermore $Z(P) = C_{p^k} \times \ker \pi$, and $P/Z(P) = V$, which looks rather like the ingredients of Blackburn's classification.

So now my question again: does every finite $p$-group with derived subgroup of order $p$ arise in this way?