  For discrete groups G, we introduce equivariant Nielsen invariants. They are
equivariant analogs of the Nielsen number and give lower bounds for the number
of fixed point orbits in the G-homotopy class of an equivariant endomorphism
f:X->X. Under mild hypotheses, these lower bounds are sharp.
  We use the equivariant Nielsen invariants to show that a G-equivariant
endomorphism f is G-homotopic to a fixed point free G-map if the generalized
equivariant Lefschetz invariant of f is zero. Finally, we prove a converse of
the equivariant Lefschetz fixed point theorem.
