Free Actions of Finite Groups on $S^n\times S^n$

Let $p$ be an odd prime. We construct a non-abelian extension $\mathrm{\Gamma }$ of $S^1$ by $Z\mathrm{/}p\times Z\mathrm{/}p$, and prove that any finite subgroup of $\mathrm{\Gamma }$ 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.