Constructions of exotic actions on product manifolds with an asymmetric factor

We explore transformation groups of manifolds of the form $M\times S^n$, where $M$ is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group. In particular, we prove that for $n=2$ there exists an infinite family of distinct non-diagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on $M \times S^1$, where $M$ belongs to the class of "almost asymmetric" manifolds considered previously by V. Puppe and M. Kreck.