Group algebras with almost maximal Lie nilpotency index

Let $KG$ be a non-commutative Lie nilpotent group algebra of a group $G$ over a field $K$. It is known that the Lie nilpotency index of $KG$ is at most $|G'|+1$, where $|G'|$ is the order of the commutator subgroup of $G$. In [V. Bovdi, E. Spinelli: \emph{Modular group algebras with maximal Lie nilpotency indices}, Publ. Math. Debrecen, 65 (2004), 243-252] the groups $G$ for which this index is maximal were determined. Here we list the $G$'s for which it assumes the next highest possible value