a free product action on a product of spheres. For example, it is known that the alternating group $A_4$ cannot act freely on a product of spheres by a product action. This follows from a result of Oliver [@oliver Theorem 1] which says that $A_4$ does not act freely on any product of equal dimensional spheres with trivial action on homology. On the other hand, it can be shown that $A_4$ acts freely on ${\mathbb S}^2 \times {\mathbb S}^3$ with trivial action on homology. So, one cannot answer the above question affirmatively by considering only the product actions.

The situation with $A_4$ is not an exceptional case. For example, if one searches through the characters of a finite group $G$ and tries to see when $G$ has a family of characters $\chi _1, \dots, \chi _k$ such that $G$ acts freely on ${\mathbb S}(\chi _1) \times \cdots \times {\mathbb S}(\chi _k )$, then one notices that not many groups have such a family of characters. In fact, Urmie Ray [@ray] showed that finite groups which have free linear actions are very rare: If a finite group $G$ has a free linear action on a product of spheres, then all nonabelian 