  Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the
non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. We prove that if
$G$ is a finite potent $p$-group, then $[G,G^{\varphi}]$ and the $k$-th term of
the lower central series $\gamma_k(\nu(G))$ are potently embedded in $\nu(G)$
(Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the
exponent $\exp(\nu(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study
the weak commutativity construction of powerful $p$-groups (Theorem C).
