The group class SJ is not countably recognizable

A group is an {\it $\text{SJ}$-group} if it admits a general abelian series all of whose terms are subnormal of finite defect in the whole group. We construct a group $G$ which is not an~\hbox{$\text{SJ}$-group}, although every countable subgroup of $G$ is an $\text{SJ}$-group.