  We show that if $\mathbf H$ is a Fitting pseudovariety of groups and $\mathbf
V$ is a local pseudovariety of monoids, then $\mathbf {LH}m \mathbf V$ is local
if either $\mathbf V$ contains the six element Brandt monoid, or $\mathbf H$ is
a non-trivial pseudovariety of groups closed under extension.
