Free actions of $p$-groups on products of lens spaces

Let $p$ be an odd prime number. We prove that if $(\mathbf{Z}/p)^r$ acts freely on a product of $k$ equidimensional lens spaces, then $r\leq k$. This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian $p$-groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A $p$-group can act freely on a product of $k$lens spaces with a trivial action on homology if and only if $\operatorname{rk}(G)\leq k$ and $G$ has the $\Omega$-extension property. The main technique is to study group extensions associated to free actions.