  A polyhedral product is a natural subspace of a Cartesian product, which is
specified by a simplicial complex K. The automorphism group Aut(K) of K induces
a group action on the polyhedral product. In this paper we study this group
action and give a formula for the fixed point set of the polyhedral product for
any subgroup H of Aut(K). We use the fixed point data to compute examples of
Bredon cohomolohgy for small non-Abelian groups such as $D_8$ and $\Sigma_4$.
