Semigroups arising from families of normal subgroups

Given any family of normal subgroups of a group, we construct in a natural way a certain monoid, the group of units of which is a semidirect product. We apply this to obtain a description of both the semigroup of endomorphisms and the group of automorphisms of an Ockham algebra of finite boolean type. We also determine when such a monoid is regular, orthodox, or inverse.