Equivariant deformations of manifolds and real representations

In the paper we give a partial answer to the following question: Let G be finite group acting smoothly on a compact ( smooth) manifold M, such that for each isotropy subgroup H of G the submanifold M-H fixed by H can be deformed without fixed points; is it true that then M can be deformed without fixed points G-equivariantly? The answer is no, in general. It is yes, for any G-manifold, if and only if G is the direct product of a 2-group and an odd-order group.