Induced representation from subgroup to subgroup I wonder why the third of the properties of the induced representation here (http://mathworld.wolfram.com/InducedRepresentation.html) holds. Does it follow from the universal property? I could not find the question on stackexchange.
Coming from the definition (3) as in Facts about induced representations
 A: Edit: somehow I was tired and misread the question, i.e. I answered a different one. But hey, the thing I answered to begin with is cool, so let's just leave it there as extra. When $K \le H \le G$ are subgroups,
$$
k[G] \otimes_{k[H]} k[H] \otimes_{k[K]} V = k[G] \otimes_{k[K]} V
$$
gives you the induction formula. 
I noticed they defined the action incorrectly, which makes it very hard to understand what's going on. The correct definition is written down here in the topological group and unitary representations case, or in this post in the finite group case over an arbitrary field.
Let $k$ be a field (a priori, it could also be a ring, but I want to keep saying words like "vector bundle"). In this post, I consider a finite group $G$ with subgroup $H$ and a representation $\rho : H \to \mathrm{GL}(W)$. We define a group action on $G \times W$ via $\bar g \cdot (g,w) \overset{def}= (\bar g g,w)$ and the equivalence relation
$$
\forall h \in H, \quad (g,w) \sim (gh, \rho_h^{-1}(w)). 
$$
(This equivalence relation mimics the property of the tensor product $k[G] \otimes_{k[H]} W$ where $gh \otimes w = g \otimes \rho_h(w)$). Denote the equivalence class of $(g,w)$ by $[g,w]$. This equivalence relation is $G$-invariant :
$$
\bar g \cdot (gh,\rho_h^{-1}(w)) = (\bar g gh,\rho_h^{-1}(w)) \sim (\bar gg, w),
$$
so that letting $\mathcal L_W \overset{def}= G \times W / \!\sim$, we obtain an action of $G$ on $\mathcal L_W$ (via the above formula, i.e. $\bar g \cdot [g,w] \overset{def}= [\bar gg,w]$). We also have a projection map of $G$-sets given by $\pi : \mathcal L_W \to G/H$ defined by $\pi([g,w]) \overset{def}= gH$. This turns $\mathcal L_W$ into a vector bundle (operations performed pointwise) over $G/H$ with fiber $W$ since choosing a system of left coset representatives $\{r_1,\cdots,r_{|G|/|H|}\}$,  we can produce an isomorphism $\mathcal L_W \simeq G/H \times W$ of vector bundles over $G/H$ via $[r_i,w] \mapsto (r_i,w)$. However, this is not convenient to study the action of $G$ on $\mathcal L_W$. 
Let $\Gamma(G/H,\mathcal L_W)$ denote the space of sections of this vector bundle, namely the set of functions $s : G/H \to \mathcal L_W$ written $s(gH) = [g,s_g]$ where $s_g \in W$ (the sections are not required to be maps of $G$-sets). Since $[g,s_g] = [gh,\rho_h^{-1}s_g]$, the section $s \in \Gamma(G/H,\mathcal L_W)$ corresponds to a function $\bar s : G \to W$ satisfying $\bar s_{gh} = \rho_h^{-1} \bar s_g$. The correspondence $s \mapsto \bar s$ is $k$-linear and we will put an action of $G$ on both $\Gamma(G/H, \mathcal L_W)$ and
$$
\mathrm{Ind}_H^G(W) \overset{def}= \{ \bar s : G \to W \mid \forall h \in H, \, g \in G, \quad \bar s_{gh} = \rho_h^{-1} \bar s_g\}
$$
which makes this correspondence $k[G]$-linear. 
Consider the map $\Gamma(G/H,\mathcal L_W) \to k[G] \otimes_{k[H]}W$ defined by 
$$
\left( s : G/H \to \mathcal L_W, s(gH) = [g,s_g] \right) \mapsto \sum_{g \in G} g \otimes s_g. 
$$
In the reverse direction, for a function $s : G \to W$, set
$$
\widehat s : G \to W, \quad \widehat s(g) = \frac 1{|H|} \sum_{h \in H} \rho_h(s(gh)).
$$
Note that $\widehat s$ satisfies the property $\rho_h(\widehat s(gh)) = \widehat s(g)$ for all $h \in H$, hence $\widehat s \in \mathrm{Ind}_H^G(W)$. We can re-write an arbitrary element of $k[G] \otimes_{k[H]} W$ as
$$
\sum_{g \in G} g \otimes s_g = \sum_{i=1}^{|G:H|} \sum_{h \in H} r_i h \otimes s_{r_ih} = \sum_{i=1}^{|G:H|} r_i \otimes \left( \sum_{h \in H} \rho_h(s_{r_ih}) \right) = |H| \sum_{i=1}^{|G:H|} r_i \otimes \widehat s(r_i). \\
\sum_{h \in H} \sum_{i=1}^{|G:H|} r_i \otimes \rho_h(\widehat s(r_ih)) = \sum_{h \in H} \sum_{i=1}^{|G:H|} r_ih \otimes \widehat s(r_ih) 
=\sum_{g \in G} g \otimes \widehat s(g). 
$$
Define the inverse map $k[G] \otimes_{k[H]}W \to \Gamma(G/H,\mathcal L_W)$ by : 
$$
\sum_{g \in G} g \otimes s_g \mapsto \left( gH \mapsto [g, \widehat s(g)] \right).
$$
This is easily seen to be a $k$-linear isomorphism, hence there is a unique choice of $k[G]$-module structure on $\Gamma(G/H,\mathcal L_W)$ which makes this a $k[G]$-linear isomorphism (i.e. "transport de structure"), namely the one given by the natural action of $G$ on $\mathcal L_W$ :
$$
\bar g \cdot (gH \mapsto [g,s_g]) \overset{def}=(gH \mapsto [g,s_{\bar g^{-1}g}]) = (gH \mapsto [\bar g g, s_g]).  
$$
What is nice about this action is that any section $s : G/H \to \mathcal L_W$ is automatically a map of $G$-sets. It also gives $\mathrm{Ind}_H^G(W)$ a $k[G]$-module structure via 
$$
(s : G \to W) \mapsto (\bar g \cdot s : G \to W, \quad g \mapsto s(\bar  g^{-1}g)).
$$
All these correspondences are natural $k[G]$-linear isomorphisms, so I hope you understand the situation better!
Hope that helps,
