Let $p$ be a prime number and $n$ an integer such that $p\geq n$. Let $P(n)$ denotes the number of partitions of $n$. Can we conclude from Theorem 1.1 and Theorem 1.3 in the reference split extension, that the number of isomorphism classes of splits extensions of $( \mathbb{Z} / p \mathbb{Z} )^n$ by $\mathbb{Z} / p \mathbb{Z} $ with a non abelian middle groups is $P(n)-1$ ?. Does the subgroup $( \mathbb{Z} / p \mathbb{Z} )^n$ should be here characteristic in $(\mathbb{Z}/p\mathbb{Z})^{n}\rtimes \mathbb{Z}/p\mathbb{Z}$ ?.

Any help would be appreciated so much. Thank you all.