In the paper we give a partial answer to the following question: Let $G$ be a 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.