A structural approach to the locality of pseudovarieties of the form   $\mathbf{LH} m \mathbf{V}$
  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. 
Benjamin Steinberg