I have been dealing with certain subgroups of group cohomology, and the following general question comes to my mind. Suppose $C$ is a chain complex of $R$-modules and $H_n(C)$ its $n$-th homology module. Now let M be an $R$-submodule of $H_n(C)$. Does there exist a chain complex $D$ of $R$-modules such that $M=H_n(D)$?
Edit: Suppose that there is a sequence of $R$-submodules $M_n$ of $H_n(C)$. When can we find a chain complex $D$ such that $H_n(D)=M_n$ for all $n$?
For any sequence of $R$-modules $M_n$ whatsoever (regardless of whether you obtained them as submodules of $H_n(C)$ for some $C$), the chain complex $D$ defined by $$\cdots\xrightarrow{\;d_{n+2}\;} M_{n+1}\xrightarrow{\;d_{n+1}\;}M_n\xrightarrow{\;d_{n}\;} M_{n-1}\xrightarrow{\;d_{n-1}\;}\cdots$$ where each $M_n$ is in the $n$th spot, and the maps $d_n$ are all the trivial homomorphisms (sending all of $M_n$ to the zero element of $M_{n-1}$) will have $$H_n(D)=\ker(d_n)/\mathrm{im}(d_{n+1})=M_n/0\cong M_n.$$
