It is well-known that a finite group is nilpotent if and only if every maximal subgroup is normal. Furthermore every nilpotent group has all maximal subgroups normal.
Question: Let $G$ be a group whose maximal subgroups are normal in $G$. If $M$ is a maximal subgroup of $G$, then are all the maximal subgroups of $M$ normal in $M$.
Of course for finite groups this is true since any subgroup of a nilpotent group is nilpotent, so my question is really about non-finite groups.
In order to get some idea of whether or not this claim is true it would be very helpful to look at some examples of non-nilpotent groups (admitting maximal subgroups) in which every maximal subgroup is normal. If someone knows of relevant literature it would be appreciated.