We work in the smooth category: manifolds and maps are meant to be smooth. Let G be a finite group acting on a connected closed manifold X and f an equivariant self-map on X with flA fixpointfree, where A is a closed invariant submanifold of X with codim A >- 3. The purpose of this paper is to give a proof using obstruction theory of the following fact: If Xis simply connected and the action of G on X- A is free, then f is equivariantly deformable rel. A to fixed point free map if and only if the usual Lefschetz number L(fl (x A» = 0. As a consequence we obtain a special case of a theorem of Wilczynski (cf . [12, Theorem A] ~. Finally, motivated by Wilczynski's paper we present an interesting question concerning the equivariant version of the converse of the Lefschetz fixed point theorem.