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

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.

ADDITION As pointed out by Noam in the comments, the above is not quite satisfying. The simplest reference is one to the paper of Kuribayashi and Kimura

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.

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.

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