  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 (or lower)
Lie nilpotency index is at most |G'|+1, where |G'| is the order of the
commutator subgroup. The authors have previously determined the groups G for
which this index is maximal and here they determine the G for which it is
`almost maximal', that is the next highest possible value, namely |G'|-p+2.
