  In this paper we continue the study of algebraic fundamentale group of
minimal surfaces of general type S satisfying K_S^2<3\chi(S). We show that, if
K_S^2= 3\chi(S)-1 and the algebraic fundamental group of S has order 8, then S
is a Campedelli surface.
  In view of the results of math.AG/0512483 and math.AG/0605733, this implies
that the fundamental group of a surface with K^2<3\chi that has no irregular
etale cover has order at most 9, and if it has order 8 or 9, then S is a
Campedelli surface.
  To obtain this result we establish some classification results for minimal
surfaces of general type such that K^2=3p_g-5 and such that the canonical map
is a birational morphism. We also study rational surfaces with a Z_2^3-action.
