Proving some prooperty in finite p-groups when $\theta_g$ is inner automorphism induced by element $g$

Question

Let $G$ be a finite p-group $(p\ge3)$ and $\theta_g$ be the inner automorphism induced by element $g$.
If $[G,g]\le Z_2(G)$ (upper central series) and $Z_2(G)$ is an abelian non-cyclic group of order $p^3$ is this enough to get
1. $\gamma_2G^p\le Z_2(G)$ (lower central series)
2. $g\in Z_2(G)$
3. $g\in Z_3(G)$
I know if (2) is right then (3) is right too I wrote both because I'm not sure (2) is right
$[G,g]=\langle [x,g]|x\in G\rangle$

Answer 1

Consider the following matrix group, upper traingular group, over $\mathbb{Z}_p$ (integers modulo $p$). $$ \begin{bmatrix} 1 & * & * & \cdots & * & *\\ 0 & 1 & * & \cdots & \cdots & * \\ 0 & 0 & 1 & * & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & 0 & 1 & * \\ 0 & 0 & \cdots & 0 & 0 & 1 \\ \end{bmatrix} $$ Here $Z_2(G)$ is "top-right triangle" $$ \begin{bmatrix} 1 & 0 & \cdots & 0 & * & *\\ 0 & 1 & 0 & \cdots & 0 & * \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & 0 & 1 & 0 \\ 0 & 0 & \cdots & 0 & 0 & 1 \\ \end{bmatrix} $$ Take for $g$ the following type element: $$ g=\begin{bmatrix} 1 & 0 & \cdots & \alpha & * & *\\ 0 & 1 & 0 & \cdots & \beta & * \\ 0 & 0 & 1 & 0 & \cdots & \gamma \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \cdots & 0 & 1 & 0 \\ 0 & 0 & \cdots & 0 & 0 & 1 \\ \end{bmatrix} $$ Here one of $\alpha,\beta,\gamma$ is non-zero, $*$ at top-right corner can be anything; remaining entries are zero. Then taking matrix size arbitrarily, but at least $4$, we can check that $g$ and $Z_2(G)$ satisfy conditions in your theorem.

However, take matrix size very large so that $G$ will contain element of large prime power order, and so $G^p$ will also contain element of large prime power order. I don't believe then that $\gamma_2G^p\leq Z_2(G)$ will hold.