Action of subgroups of the mapping class group on Heisenberg homologies

In previous work we constructed twisted representations of mapping class groups of surfaces, depending on a choice of representation $V$ of the Heisenberg group $\mathcal{H}$. For certain $V$ we were able to untwist these mapping class group representations. Here, we study the restrictions of our twisted representations to different subgroups of the mapping class group. In particular, we prove that these representations may be untwisted on the Torelli group for any given representation $V$ of $\mathcal{H}$. When $V$ is the Schr\"odinger representation, we also construct untwisted representations of subgroups defined as kernels of crossed homomorphisms studied by Earle and Morita.