Suppose we have a finite group $G$ with subgroup $H\leq G$. If $R$ is a commutative ring then we have the group rings $B=R[G]$ and $A=R[H]$, along with the natural inclusion $i:A\hookrightarrow B$. From this we can induce the functors:
$\bullet$ Restriction of scalars $\mathcal{R}_H^G$ which takes a $B$-module $M$ to an $A$-module $\mathcal{R}_H^G(M)$ via the action $m\cdot a=m\cdot i(a)$; and
$\bullet$ Extension of scalars $\mathcal{E}_H^G$ which takes an $A$-module $N$ to a $B$-module $\mathcal{E}^G_H(N)=N\otimes_AB$.
Now, we can say a few obvious things about $\mathcal{R}_H^G$ and $\mathcal{E}_H^G$. Firstly $\mathcal{R}_H^G$ and $\mathcal{E}_H^G$ take free (and hence projective) modules to free (proj.) modules. They are also additive and exact. However, beyond this I am having trouble seeing an intuition for what is going on. Is there any `nice' way of viewing what is going on? Such as how to predict what $\mathcal{R}_H^G(M)$ or $\mathcal{R}_H^G(N)$ will be?
In particular, let us take (for example) $G=D_{2p}$, the dihedral group of order $2p$ ($p$ an odd prime), and $H=C_p$ the cyclic group of order $p$. Then surely $\mathcal{R}_H^G(A)=A$ since $A$ (as a $B$-mod) is simply the $B$-module in which the generator of order 2 acts trivially. Now, is this true for any finite $G$ with subgroup $H$?
These are exactly the restriction and induction functors, usually written $ \operatorname{Res}^G_H M $ and $ \operatorname{Ind}^G_H N = k[G] \otimes_{k[H]} N $. They are an adjoint pair of functors: this is a classic result called Frobenius Reciprocity. The precise statement is $$ \operatorname{Hom}_{k[G]}(\operatorname{Ind}^G_H N, M) \cong \operatorname{Hom}_{k[H]}(N, \operatorname{Res}^G_H M)$$
Over $\mathbb{C}$, you can fairly easily see from the tensor definition that $\operatorname{Ind}^G_H N$ a natural basis given by $\{g_i \otimes v_j\}$, where the $v_j$ form a basis of $N$ and the $g_i$ are coset representatives of $H$ in $G$, and so $$\dim \operatorname{Ind}^G_H N = \frac{|G|}{|H|} \dim N$$
You could predict what the restriction will be using character tables, or directly seeing it in the representation, and use Frobenius reciprocity to translate statements about restriction into statements about induction, and vice-versa.