Modular group algebras with almost maximal Lie nilpotency indices

Let K be a field of positive characteristic p and KG the group algebra of a group G . It is known that, if KG is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most |G'|+1 where |G'| is the order of the commutator subgroup. The authors previously determined those groups G for which this index is maximal and here they determine the groups G for which it is 'almost maximal', that is, it takes the next highest possible value, namely |G'|-p+2.