It is well known that a subgroup of the semidirect product $H\rtimes K$ is not in general semidirect product of two subgroups $H'\le H$ and $K'\le K$ but always exist some subgroups of $H\rtimes K$ on the form $H'\rtimes K'$.
Definition: A subgroup $G$ of $H\rtimes K$ is called split if $G\simeq (G\cap H)\rtimes (G\cap K)$.
Question 1: Let $E=(\mathbb{Z}/p\mathbb{Z})^{n}\rtimes (\mathbb{Z}/p\mathbb{Z})^{m}$ be the semidirect product of $(\mathbb{Z}/p\mathbb{Z})^{n}$ by $(\mathbb{Z}/p\mathbb{Z})^{m}$. What is the number of split subgroups of order $p^{s}$ in $E$.
Question 2: Let $E=(\mathbb{Z}/p\mathbb{Z})^{n}\rtimes (\mathbb{Z}/p\mathbb{Z})^{m}$ be the semidirect product of $(\mathbb{Z}/p\mathbb{Z})^{n}$ by $(\mathbb{Z}/p\mathbb{Z})^{m}$. What is the number of elementary abelian subgroups of order $p^{s}$ in $E$.
Any help would be appreciated so much. Thank you all.
There is a fundamental problem with your question. You seem to be under the impression that your groups $E$ are well-defined, but that is not so. For example, there are two semi-direct products of type $C_2^2 \rtimes C_2$. One is the trivial (i.e. direct) product $C_2 \times C_2 \times C_2$ and the other is $D_8$, the dihedral group of order $8$.
Ok, so maybe you want to exclude direct products. Even then you still need to specify a non-trivial action for "the" semi-direct product to make sense. For example, if $p=2$, $n=4$ and $m=1$ there are (at least) two genuine semi-direct products of type $C_2^4 \rtimes C_2$. One is the wreath product $C_2^2 \wr C_2$ and the other is $C_2^2 \times D_8$. I am not completely sure, but I think $32$ is the smallest order where there are at least two genuine semi-direct products of the type you want.