  Unitary designs are essential tools in several quantum information protocols.
Similarly to other design concepts, unitary designs are mainly used to
facilitate averaging over a relevant space, in this case, the unitary group
$\mathrm{U}(d)$. While it is known that exact unitary $t$-designs exist for any
degree $t$ and dimension $d$, the most appealing type of designs, group designs
(in which the elements of the design form a group), can provide at most
$3$-designs. Moreover, even group $2$-designs can only exist in limited
dimensions. In this paper, we present novel construction methods for creating
exact generalized group designs based on the representation theory of the
unitary group and its finite subgroups that overcome the $4$-design-barrier of
unitary group designs. Furthermore, a construction is presented for creating
generalized group $2$-designs in arbitrary dimensions.
