We investigate finite transformation groups of manifolds of the form M×Sn, where M is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group. The obvious actions are "product", or "diagonal", that is group elements act by identity on M and by some action on Sn. We construct (for low n) exotic actions on M×Sn which are not equivalent to product actions.

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. On the other hand, if M is one of the "almost asymmetric" manifolds considered previously by V. Puppe and M. Kreck, then every free circle action on M×S1 turns out to be equivalent to a diagonal one.