Defining induced representation with motivation In the book Fourier Analysis on Finite Groups and Applications
By Audrey Terras, the author starts chapter on induced representations by following quotes:



Then he goes to define induced representations, defines it as below, and raises a question to check that the given map is representation. I didn't understand this immediate travelling in a dark-room. Can one give any motivation for this definition of induced representation.



 A: I think this is a great question, and one that every expositor of representation theory has to think about!
I like motivating induction by first motivating induction of characters. Here is how this goes. Let $G$ be a finite group, $H$ a subgroup, and let ${\rm R}(G)$ denote the ring of class functions on $G$, and similarly for $H$. Both ${\rm R}(G)$ and ${\rm R}(H)$ are Hermitian inner product spaces, with inner products $\langle \cdot,\cdot\rangle_G$, respectively $\langle\cdot,\cdot \rangle_H$. Recall that e.g. $\langle \cdot,\cdot\rangle_G$ is defined by $\langle \psi,\phi\rangle_G=\frac{1}{\#G}\sum_{g\in G}\psi(g)\bar{\phi}(g)$ for $\psi$, $\chi\in {\rm R}(G)$.
Now, there is an obvious linear map ${\rm Res}\colon {\rm R}(G)\to {\rm R}(H)$ given by restriction of class functions. From general linear algebra, you know that linear maps between Hermitian inner product spaces have adjoints, i.e. there exists a unique function ${\rm R}(H)\to {\rm R}(G)$, call it (spoiler alert) ${\rm Ind}$, satisfying
$$
\langle{\rm Res}(\psi),\chi\rangle_H = \langle\psi,{\rm Ind}(\chi)\rangle_G
\quad \text{for all $\psi\in {\rm R}(G)$ and all $\chi\in{\rm R}(H)$.}
$$
Moreover, the above formula allows you to actually write down the familiar formula for induction, because the class function ${\rm Ind}(\chi)$ is uniquely determined by its inner products with all elements of a basis of ${\rm R}(G)$. Notice how, with this approach, Frobenius reciprocity becomes not a theorem but a defining property of induction.
Ok, but now you have a major question left. Inside ${\rm R}(H)$ you have these distinguished elements, namely characters of representations of $H$, and similarly for $G$. Moreover, you know that the restriction of a character is a character, since restriction is an obvious operation on representations, too. It follows from this and from Frobenius reciprocity that if $\chi$ is a character of $H$, then the inner product of ${\rm Ind}(\chi)$ with every irreducible character of $G$ is a non-negative integer, which implies that ${\rm Ind}(\chi)$ is also a character. So a natural question is: can you write down a representation whose character is ${\rm Ind}(\chi)$? At this point, you stare at the formula for the class function  ${\rm Ind}(\chi)$ that we obtained from the adjointness property, you wait for a stroke of divine inspiration, and you write down the representation that you see in the book.
I do not actually know whether this is how Frobenius discovered this. But to be honest, Frobenius's work from that period is full of things that make me scream "How on earth does anybody come up with that?!".
