  In this paper, we construct an infinite family of hemisystems of the
Hermitian surface $\mathsf{H}(3,q^2)$. In particular, we show that for every
odd prime power $q$ congruent to $3$ modulo $4$, there exists a hemisystem of
$\mathsf{H}(3,q^2)$ admitting $C_{(q^3+1)/4} : C_3$.
