  The distinguishing number of $G \leqslant \sym(\Omega)$ is the smallest size
of a partition of $\Omega$ such that only the identity of $G$ fixes all the
parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and
Seress on the distinguishing number of finite primitive groups, we show that
all imprimitive quasiprimitive groups have distinguishing number two, and all
non-quasiprimitive semiprimitive groups have distinguishing number two, except
for $\mathrm{GL}(2, 3)$ acting on the eight non-zero vectors of $\mathbb
F_2^3$, which has distinguishing number three.
