If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition $\lambda=\rho\circ\iota:H\to GL(V)$ where $\iota:H\to G$ is the inclusion map. Now if I let $V^{H}\subset V$ be the subspace of $V$ fixed by $H$ (the image of the projection map $\frac{1}{|H|}\sum_{h\in H}\lambda(h)$) then what can I say about the dimension of $V^{H}$? Can it be understood in terms of the dimension of $V$, the order of $H$, and the order of $G$?
This seems intimately related to the following question. Are there versions of the orbit-stabilizer theorem and/or Burnside lemma in which dimensions (rather than cardinalities) appear and that can be applied in the theory of linear representations?
I'm particularly interested in the case that $(\rho,V)$ is the regular representation modulo the trivial representation.
As to the broader context of this question. Aside from being a purely academic question that I'd like to know the answer to, I was motivated to ask it when attempting to understand an answer provided to the following question on MathOverlow: