This is a question about Hirshon's paper "The center and the commutator subgroup in hopfian groups". In Theorem 12, he showed that if $B$ is a perfect group and $A$ is a hopfian group, then the direct product $A \times B$ is hopfian. Does $B$ have to be a group with finitely many normal subgroups?

This is a question about Hirshon's paper "The center and the commutator subgroup in hopfian groups"(https://projecteuclid.org/euclid.afm/1485894451). Theorem 12 stating that if $B$ is a perfect group with finitely many normal subgroups and $A$ is a hopfian group, then the direct product $A \times B$ is hopfian. Is it possible to weaken the "finitely many normal subgroups" hypothesis?