The Specht modules are of fundamental importance to the representation theory of the symmetric group, and their 0th cohomology is understood through entirely combinatorial methods due to Gordon James. Over fields of odd characteristic, Hemmer proposed a similar combinatorial approach to calculating their 1st degree cohomology, or extensions by the trivial module. This combinatorial approach motivates the definition of universal $p$-ary designs, which we shall classify. We then explore the consequences of this classification to problem of determining extensions of Specht modules. In particular, we classify all extensions of Specht modules indexed by two-part partitions by the trivial module and shall see some far-reaching conditions on when the first cohomology of a Specht module is trivial.
