On the graph of induction-restriction for group-subgroup representations Let $G$ be a finite group, and $H$ a subgroup.
Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ and $H$ (up to isom.).    
Consider the graph $\mathcal{G}(H \subset G )$ whose vertices are $\{ V_i, W_j : i \in I, j \in J   \}$ such that there are $d_{ij}$ edges between $V_i$ and  $W_j$ if $\langle V_i\vert_H,W_j \rangle = d_{ij}$.   
Let $\mathcal{G}_0(H \subset G )$ be the connected component of the trivial representation $V_0$ (noted $\star$)  of $G$.
Note that if $K \subset H$ is a normal subgroup of $G$ then $\mathcal{G}_0(H \subset G )= \mathcal{G}_0(H/K \subset G/K )$.  
The graph $\mathcal{G}_0$ is $n$-supertransitive ($n$-$st$) if up to distance $n$ from $\star$  it's just a linear chain:  $$\star-\circ-\circ-\circ-\circ-\circ-\circ-\circ-\circ$$
Let $st(H \subset G )$ be the integer $n$ such that the graph $\mathcal{G}_0(H \subset G )$ is $n$-$st$ but not $(n+1)$-$st$.
Note that $st(H \subset G ) \ge 1$  and $st(H \subset G )= st(H/K \subset G/K )$.    
Examples: $st(\{e\} \subset \mathbb{Z}_2)=2$ and $st(\{e\} \subset G)=1$ if $G \neq \mathbb{Z}_2$.
$st(\mathbb{Z}_2 \subset D_{10})=1$ and $st(S_4 \subset S_5)=3$  

Question: Is there $(H_0 \subset G_0 )$ with $st(H_0 \subset G_0 ) > 7$ ? Is $st(H \subset G )$ bounded ? 


Optional questions: Is there usual names for $\mathcal{G}_0(H \subset G )$ and $st(H \subset G )$ in groups theory ?  References ?  Is there a   program drawing $\mathcal{G}_0(H \subset G )$ and computing $st(H \subset G )$ ? 
 A: (This is just a comment with some ideas and calculations, and graphics in comments are hard.)

For $G=S_7$, $H=S_6$, the perm rep $1_H^G$ is $7$-transitive, but only 3 supertransitive. The graph is connected, so $\mathcal{G}=\mathcal{G}_0$.
The graph is a bit weird looking. All edges are single edges. $$\begin{array}{}
& & && \hspace{-2ex}\circ \to \circ \to \circ \\
& & & \hspace{-2ex}\circ \to \circ \rlap{\rightarrow}^{\nearrow} & \hspace{-2ex} \circ\\
      & \hspace{-2ex}\circ\to\circ \rlap{\rightarrow}_{\searrow} &\hspace{-1.5ex}\circ \to \circ \rlap{\rightarrow}^{\nearrow} & \hspace{-2ex}\circ \to \circ \rlap{\rightarrow}^{\nearrow} & \hspace{-2ex} \circ\\
\star \to \circ \to \circ \to \circ {}^{\nearrow}_{\searrow} {} && \hspace{-1.5ex}\circ  \to \circ \rlap{\rightarrow}_{\searrow}^{\nearrow} & \hspace{-2ex}\circ \to \circ {}^\nearrow\\
      & \hspace{-2ex}\circ \to\circ \rlap{\rightarrow}^{\nearrow} &\hspace{-1.5ex}\circ \to \circ \to &\hspace{-2ex}\circ\\
\end{array}$$

By analogy with modular representation theory, I'd call the (bipartite) graph a “decomposition matrix”, and the connected components “blocks”, with $\mathcal{G}_0$ the “principal block.” I suspect it has been studied for a long time.
It can be computed as:
mat := MatScalarProducts(Irr(h),RestrictedClassFunctions(Irr(g),h));

in GAP. I found it helpful to write out the edge relation:
edges := Filtered( Cartesian([1..Size(mat)],-[1..Size(mat[1])]),
  ij -> not IsZero(mat[ij[1]][-ij[2]]));

Probably a lot can be said about this matrix, but I haven't had time to think about it.

Sticking with the definition of $st(H \leq G)$, I find that for $|G| \leq 120$, $st(H \leq G) < 3$ except:


*

*$st(S_2 \leq S_3) = 4$

*$st(S_3 \leq S_4) = 3$

*$st(A_4 \leq A_5) = 3$

*$st(S_4 \leq S_5) = 3$

*$st(AGL(1,5) \leq PGL(2,5)) = 3$ (S5 = PGL(2,5))


Then checking all $H\leq G$ with $[G:H] \leq 15$ we see the same sorts of groups (natural Sn, natural An, some variants of natural PGL2, natural Mathieu groups) but also some weird ones (AGL, unnnatural mathieu). Also, other than natural $S_3$, $st(H \leq G) \leq 3$.
A: 
Answer (due to Dave Penneys):  $st(H\subset G) \le 3$ if $[G:H]>3$.

I report here his comment :  

A subgroup subfactor (of index $> 3$) can be at most $3$-supertransitive. The $4$-box
  space is at least 15 dimensional. This can be seen by looking at the
  partition planar algebras (MR2972458).  

As observed by Jack Schmidt, if $st(H\subset G) > 3$ and $[G:H] \le 3$ then $(H\subset G) \sim (S_2 \subset S_3 )$ and $st(S_2 \subset S_3 ) = 4$.         
By reading the pages 13-14 of the paper cited by Dave (also with this book ex. 4.2.3 p53 and prop. A.4.4 p141), we see that "for $k=1,2$ or $3$", $st(H \subset G) = k$ iff $1_H^G$ is (at least) $k$-transitive, iff $G/H_G$ is (at least) a $k$-transitive group (it's false for $k \ge 4$).   
Then the classification (up to $\sim$ ) of the inclusions $(H \subset G)$ with $st(H \subset G) = 3$ is given by the classification of the $3$-transitive groups.  
Classification of the $4$-transitive groups: Symmetric, Alternating and Mathieu groups only (here).   

Questions: Is there a classification of the $3$-transitive groups ? A conjecture ?
  Is there a table of the number of $3$-transitive groups of degree $n$ ?

