No, this cannot happen. Suppose $C$ is a $(p,q)$ complete intersection. The line bundle $\mathscr{O}_C(1)$ induced by $\mathscr{O}_{\mathbb{P}^3}(1)$ is the only line bundle $L$ on $C$ with $L^{p+q-4}=K_C$ and $h^0(L)=4$; therefore any automorphism $u$ of $C$ comes from an automorphism $\tilde{u} $ of $\mathbb{P}^3$. The fixed locus of $\tilde{u} $ is a union of linear subspaces; it follows easily that $u$ has at most $pq+1$ fixed points. Then the Riemann-Hurwitz formula implies that the genus $g$ of the quotient curve is large. For instance, for $p=q=4$, you get $g\geq 13$ if $u^2=\operatorname{Id} $, and $g\geq 9$  if $u^3=\operatorname{Id} $.