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Dold manifolds 
$P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and 
$ \mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to 
${\prod_{i=1}^{k} P(2m_i,n_i)}$, then 
$G\cong (\mathbb{Z}_2)^l$ for some 
$l\leq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.
$\mathbb{Z}_{2}$-ACTIONS ON SOME DOLD MANIFOLDS
We determine the possible 
$\mathbb{Z}_{2}$-cohomology rings of orbit spaces of free actions of 
$\mathbb{Z}_{2}$ (or fixed point free involutions) on the Dold manifold 
$P(1,n)$, where 
$n$ is an odd natural number.
