Unitary t-groups

Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t\ge 2$ in any dimension $d\ge 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a unitary $t$-group for any $t\ge 6$.