  We define homotopy group actions in terms of families of $A_\infty$ algebras
indexed by a manifold M. We give explicit formulae for the $A_\infty$ morphism
induced by a path on the manifold and for the $A_\infty$ homotopy corresponding
to a pair of homotopic paths. Finally, we compute examples for finite groups
and finitely generated free nonabelian groups and determine that every homotopy
group action by a finite group is homotopic to a strict group action.
