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][1]
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.
ADDITION As pointed out by Noam in the comments, the above is not quite satisfying. The simplest reference is one to t[he paper of Kuribayashi and Kimura][2]
Akikazu Kuribayashi and Hideyuki Kimura, MR 1068416 Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, 80--103. Unfortunately, that paper's notation is somewhat hard to penetrate. On the other hand, the way they describe the automorphism group (conjugacy class in $GL()$ should make it easy to check that the relevant surface is not hyperelliptic. [1]: https://www.evernote.com/shard/s24/sh/443727bb-27cd-41ca-ba96-8aec175385d1/eb99cdf97d58f161c6a856bfba80eb11 [2]: https://www.evernote.com/l/ABhoritVc3dDkq1BL_piYVCkr7dKhoZcDdw