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In previous work we constructed twisted representations of mapping classgroups of surfaces, depending on a choice of representation $V$ of theHeisenberg group $\mathcal{H}$. For certain $V$ we were able to untwist thesemapping class group representations. Here, we study the restrictions of ourtwisted representations to different subgroups of the mapping class group. Inparticular, we prove that these representations may be untwisted on the Torelligroup for any given representation $V$ of $\mathcal{H}$. When $V$ is theSchr\"odinger representation, we also construct untwisted representations ofsubgroups defined as kernels of crossed homomorphisms studied by Earle andMorita.

Action of subgroups of the mapping class group on Heisenberg homologies