Does the number of actions a group can have on the left cosets of a subgroup of index $n$ tell us anything about the number of such subgroups? I have a question asking me to show that the number of index $n$ subgroups of a group $G$ of rank $r$ is bounded above, and it offers the hint that I should consider the number of actions $G$ can have on a specific index $n$ subgroup $H$'s left cosets.
However, I am struggling to see why this is helpful for this question, since I can't quite understand how this relates to other index $n$ subgroups.
I suspect that it will involve some sort of pigeonhole argument, but I can't immediately how.
I think regarding the number of actions that $G$ can have on the left cosets of $H$, we can consider the action of a generating set of size $r$. Specifically as we want just an upper bound (I think), we may ignore whether or not the action of the generating set actually extends appropriately to group action of $G$. Specifically, we can just pick arbitrary mappings of each left coset, for each of the $r$ generators. 
In that case for fixed generator $g_1$, it can send any of the $n$ left cosets to any others, giving a total upper bound of $n^n$ mappings. Considering then that there are $r$ generators, that we may regard as "independent", we see there are at most $n^{nr}$ total actions it can have on $G/H$. This is then an upper bound for the total number of group actions $G$ may have on the set of left cosets $G/H$.
I suppose now we consider another group of index $n$, and consider how each of it's left cosets intersect with the left cosets of $H$? I'm not really sure what I can learn about them though, since it is an arbitrary group of index $n$.
[EDIT]: As noted in the comments, the upper bound for the number of actions may be improved to $n!^r$.
 A: I should perhaps make my comments into an answer.
An action of a group $G$ on a set $X$ is determined by the action of its generators, so if $G$ is generated by $r$ elements then the number of actions is at most $n!^r$, where $|X|=n$.
Now, for any subgroup $H$ of $G$ of index $n$, there is an action of $G$ on the set of left cosets of $H$ in $G$, where $g \in G$ maps coset $xH$ to $gxH$. We can number the cosets $1,2,\ldots,n$, where $H$ is numbered $1$, giving us an action of $G$ on $X = \{1,2,\ldots,n\}$. Then, in this action,  $H$ is the stabilizer of $1 \in X$, so different subgroups define different actions.
So for every subgroup $H$ of index $n$ in $G$, there is an action of $G$ on $X$ in which $H$ is the stabilizer of $1$. So the total number of such subgroups is bounded above by the total number of actions, which is $n!^r$.
In fact every subgroup of index $n$ gives rise to many different actions in this way, because we can choose the numbering of the cosets other than $H$ any way we like. Also, not every choice of $r$ permutation sof $X$ gives rise to an action of $G$ on $X$, and only the transitive actions arise as coset actions. So this upper bound is usually very imprecise.
