This work deals with the stable representation theory of categories related to various families of symmetric groups. In particular, we study the categories FB, FI, and introduce the new category $\PD$. The notion of representation stability is recast in the setting of FB-bimodules. The first part of the present work is an application of theory of FI-modules to a family of groups $\Gamma_{n,s}$ arising in the study of free group automorphisms. We observe that the cohomology of these groups determines an FI-module $H^i(\Gamma_{n,\bullet})$ which we show is finitely generated of stability degree n and weight i. It follows that the sequence ${H^i(\Gamma_{n,s})}s$ is representation stable in the range $s \geq i +n$, an improvement on the previously known stable range. Another consequence of this finitely generated FI-module structure is the existence of character polynomials which determine the stable characters of $H^i(\Gamma{n,s})$. In particular, this implies that the dimension of $H^i(\Gamma_{n,s})$ is given by a single polynomial in s for $s \geq i+n$. We compute explicit examples of such character polynomials to demonstrate this phenomenon. Next we provide an algorithm that computes certain structural coefficients $c_{\lambda\mu}$ related to the n-th tensor power of the free associative algebra on a vector space $\mathcal{T}(V)^{\otimes n}$. By extending the known range of computation by a factor of over 750 we reveal striking patterns that motivate our recasting of representation stability to families of bimodules. Finally, we develop the theory of $\PD$-modules. Our main result is that finitely generated $\PD$-modules give rise to representation stable families of bimodules over symmetric groups. We provide two main examples of this framework. First we show that the coefficients $c_{\lambda\mu}$ determine a finitely generated $\PD$-module and thus provide a first example of this new representation stability. Second, we introduce the extended Whitney homology of the lattice of set partitions, and show that it determines a finitely generated $\PD$-module. It is known that the ordinary Whitney homology of the lattice of set partitions forms a finitely generated FI-module, and we are able to recover, and generalize this result.