  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. Previously we determined the groups G for which the
upper/lower nilpotency index is maximal or the upper nilpotency index is
`almost maximal' (that is, of the next highest possible value, namely |G'|-p
+2). Here we determine the groups for which the lower nilpotency index is
`almost maximal'.
