On cyclicity of a module

Let $$A$$ be a $$\text{ von Neumann algebra }$$, $$\mathcal{H}$$ is a cyclic $$A$$ module, $$G$$ be a finite group acting on $$A$$, is $$\mathcal{H}$$ cyclic module over fixed point subalgebra of the action? Further for which class of $$A$$-s the result is positive!!

• Did you at least consider the finite-dimensional case? Obviously not, since if we take $A = H = M_n C$ (with $n > 1$), and $G$ a finite group acting irreducibly on $A$ (lots exist), then the fixed point algebra is just $C$, so $H$ is not cyclic over the fixed point algebra. If $A$ is a type II finite factor and the fixed point algebra is itself a factor, then there is a dimension associated, and the hoped-for result is still not true. – David Handelman May 8 at 13:29
• I am not taking ergodic action, so the fixed point algebra as huge as $M$ itself. – user136400 May 9 at 9:06