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] 

<cite authors="S. Allen Broughton" mrnumber="1090743" cite="_J. Pure Appl. Algebra_ **69** (1991), no. 3, 233--270">_S. Allen Broughton_, MR 1090743 [**Classifying finite group actions on surfaces of low genus**](http://dx.doi.org/10.1016/0022-4049(91)90021-S), _J. Pure Appl. Algebra_ **69** (1991), no. 3, 233--270.</cite>

So, modulo some care, the answer seems to be YES.


  [1]: https://www.evernote.com/shard/s24/sh/443727bb-27cd-41ca-ba96-8aec175385d1/eb99cdf97d58f161c6a856bfba80eb11