Suppose that $A$ is a finite, generating subset of a group $G$, and that $H$ is a subgroup such that $A^2$ is a union of left $H$-cosets; moreover, $H$ is maximal subject to this property. Is it true that if $|A^2|<2|A|$, then $H$ is normal? (Here $A^2$ is the set of all products $aa'$ with $a,a'\in A$.)
Quite likely, there is a simple counterexample, but I couldn't find one.
