Suppose I have a group $G$ and a subgroup $H< G,$ where $|H| = k, |G| = k l.$ Let $A_H < Aut(G) = \{\phi \left| \phi(H) = H\right.\}.$ The question is: is there any decent (where you can define "decent" in your favorite way) universal upper bound on the index of $A_H$ in $Aut(G)$ in terms of $k, l?$

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`$\left(\begin{array}{c} kl \\ k\end{array}\right)$`

is decent? – Will Sawin Nov 8 '12 at 2:12