  We study the structure of the algebraic fundamental group for minimal
surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any
irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic
fundamental group of S has order at most 5, and equality only occurs if S is a
Godeaux surface. We also show that if K_S^2<= 3\chi-3 and the algebraic
fundamental group of S is not trivial, then it is Z_2, or Z_2^2 or Z_3.
Furthermore in this last case one has: 2<=\chi<=4, K^2=3\chi-3 and these
possibilities do occur.
