  In this work we study some structural properties of the group $\eta^q(G, H)$,
$q$ a non-negative integer, which is an extension of the $q$-tensor product $G
\otimes^q H)$, where $G$ and $H$ are normal subgroups of some group $L$. We
establish by simple arguments some closure properties of $\eta^q(G,H)$ when $G$
and $H$ belong to certain Schur classes. This extends similar results
concerning the case $q = 0$ found in the literature. Restricting our
considerations to the case $G = H$, we compute the $q$-tensor square $D_n
\otimes^q D_n$ for $q$ odd, where $D_n$ denotes the dihedral group of order
$2n$. Upper bounds to the exponent of $G \otimes^q G$ are also established for
nilpotent groups $G$ of class $\leq 3$, which extend to all $q \geq 0$ similar
bound found by Moravec in [21].
