  We construct supercharacter theories for a collection of unipotent matrix
groups and produce a Hopf monoid from the supercharacters. These supercharacter
theories are coarser than those defined by Diaconis--Isaacs for algebra groups
and have supercharacters and superclasses indexed by nonnesting labeled set
partitions. We compute the supercharacter tables and describe the product and
coproduct of the Hopf monoid combinatorially. We also show that this Hopf
monoid is free.
