Space of maps in induced representation One of the definition of induced representation from a subgroup $H$ to the group $G$ comes from certain types of maps; which are termed somewhere as $H$-equivariant maps.
Let $G$ be a finite group, $H$ be a subgroup acting on vector space $V$ (so $V$ is a representation of $H$).
Then, one way of getting induced representation ${\rm Ind}_H^G(V)$ is as follows:
(1) The underlying vector space for $G$ is a subspace of all maps from $G$ to $V$: it is
$$
\{ f:G\rightarrow V \,\, | \,\, f(h\cdot x)=h\cdot f(x) \,\,\, \forall h\in H, \,\, \forall x\in G\}.
$$
(2) On this space, define action of $G$ by
$$
g\cdot f \mbox{ is the map from $G$ to $V$, which takes $x$ to $f(xg)$};
$$
i.e. $(g\cdot f)(x)=f(xg)$.
Question: Why it is necessary to consider the maps to be $H$-equivariant in above sense, which, in the beginning do not spark naturally (at least to me) when guessing about induced representation? I was able to interpret the condition of $H$-equivariant maps diagrammatically as below: but I do not get, why they are necessary in induced representation?

 A: Each $H$-equivariant function $G\to V$ has all its values on $H\subset G$ determined by one value, say on $e\in H$. On $G\backslash H$ it may then do something else. Then by $(h\cdot f)(x)=h\cdot(f(x))$ you can see, that you recover your old representation "inside the new one" by restricting the functions $G\to V$ to $H$ as the space of $H$-equivariant functions on $H$ with values in $V$ is isomorphic to $V$.
Edit: Possibly it's better to view this "inside the new one" from above as follows. The space of all $H$-equivariant functions $G\to V$ contains the subspace of $H$-equivariant functions $G\to V$ which vanish on $G\backslash H$. This space is, by the argument above, isomorphic to $V$ and the new representation of $G$ restricted to $H$ again, acting on this subspace, is indeed equivalent to your initial representation of $H$ by means of this isomorphism. Maybe this is a better view on how the new representation contains the old one in the sense of a more or less canonical continuation.
A: Another useful way to think about this construction is about the intention of it, namely, to construct an adjoint functor to the functor $R^G_H$ which restricts $G$-repns to $H$-repns. Depending on context and whether we want a left or right adjoint, that is that we want Frobenius Reciprocity $\mathrm{Hom}_G(\mathrm{Ind}^G_HV,\,W)\approx \mathrm{Hom}_H(V,R^G_HW)$ for all $G$-repns $W$.
For finite groups, in characteristic $0$ at least, that construction permits a simple proof that it fulfills the requirement.
