  Let $G$ be a group and $q$ a non-negative integer. We denote by $\nu^q(G)$ a
certain extension of the $q$-tensor square $G \otimes^q G$ by $G \times G$. In
this paper we derive a polycyclic presentation for $G \otimes^q G$, when $G$ is
polycyclic, via its embedding into $\nu^q(G)$. Furthermore, we derive
presentations for the $q$-exterior square $G \wedge^q G$ and for the second
homology group $H_2(G, \mathbb{Z}_q).$ Additionally, we establish a criterion
for computing the $q-$exterior centre $Z_q^\wedge (G)$ of a polycyclic group
$G, $ which is helpful for deciding whether $G$ is capable modulo $q$. These
results extend to all $q \geq 0$ existing methods due to Eick and Nickel for
the case $q = 0$.
