We prove that if $G$ is a countable, discrete group having infinite normal subgroups with the relative property (T) of Kazhdan–Margulis, then the Bernoulli shift action of $G$ on $\prod_{g \in G} (X_0, \mu_0)_g$, for $(X_{0},\mu_{0})$ an arbitrary non-trivial probability space, has first cohomology group isomorphic to the character group of $G$.