There is a non-hyperelliptic (fixed-point free) involution of the surface of genus $5,$ with the quotient a surface of genus $3.$ Further, $A_4$ does come up as the automorphism group thereof, see
S. Allen Broughton, MR 1090743 Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991), no. 3, 233--270.
So, modulo some care, the answer seems to be YES.