  Let $p$ be an odd prime. We construct a non-abelian extension $\Gamma$ of
$S^1$ by $Z/p \times Z/p$, and prove that any finite subgroup of $\Gamma$ acts
freely and smoothly on $S^{2p-1} \times S^{2p-1}$. In particular, for each odd
prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank
two $p$-groups on $S^{2p-1} \times S^{2p-1}$. These results arise from a
general approach to the existence problem for finite group actions on products
of equidimensional spheres.
