  We show that the existence of a left-invariant pluriclosed Hermitian metric
on a unimodular Lie group with a left-invariant abelian complex structure
forces the group to be $2$-step nilpotent. Moreover, we prove that the
pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step
nilpotent Lie group preserves the Strominger K\"ahler-like condition.
