  It is well-known that the representation theory of the finite group of
unipotent upper-triangular matrices $U_n$ over a finite field is a wild
problem. By instead considering approximately irreducible representations
(supercharacters), one obtains a rich combinatorial theory analogous to that of
the symmetric group, where we replace partition combinatorics with
set-partitions. This paper studies the supercharacter theory of a family of
subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several
combinatorial indexing sets for the supercharacters, supercharacter formulas
for these indexing sets, and a combinatorial rule for restricting
supercharacters from one group to another. A consequence of this analysis is a
Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on
set-partitions (analogous to the corresponding symmetric group rule on
partitions).
