Maybe this is one of those obvious questions that indicates that I do not understand the definitions, but I need a bit of help.
Let $H\leq G$ and $M$ be a $H$-module, the induced module is defined as $\text{Ind}^G_H M=\mathbb{Z}G\otimes_{\mathbb{Z}H}M$, which is clearly a $G$-module. Shapiro's lemma states that: $$H_\ast(H,M)\cong H_\ast(G,\text{Ind}^G_H M)$$ Now, suppose that $M=\mathbb{F}$ is a field, which is an $H$-module with the trivial action. What does shaphiro's lemma tell us in this setting? More preciselly, is there a way to understand $\text{Ind}^G_H \mathbb{F}$?
In particular I want to know if there is a way to relate $H_\ast(H,\mathbb{F})$ and $H_\ast(G,\mathbb{F})$, which I don't know wheter it is possible or not.