Finite groups with all maximal subgroups isomorphic Suppose $G$ is a finite group such that any two maximal subgroups of $G$ are isomorphic. What can be said about such a group? Can they be classified?
The finite groups that have a unique maximal subgroup are exactly the cyclic groups of prime power order, $\mathbb{Z}/p^n\mathbb{Z}$ - these are the simplest examples of such groups. Also powers of $\mathbb{Z}/p^n\mathbb{Z}$, i.e. $(\mathbb{Z}/p^n\mathbb{Z})^m = \mathbb{Z}/p^n\mathbb{Z} \times \ldots \times \mathbb{Z}/p^n\mathbb{Z}$, have this property, and I think this covers all abelian groups with this property.
In general I think such a group has to be a $p$-group, by considering maximal subgroups containing Sylow subgroups for different primes. 
This paper https://bib.irb.hr/datoteka/402744.SiCh.pdf calls such groups isomaximal, but seems to only handle $2$-groups up to order $64$.
Further question: what about groups $G$ such that any two maximal subgroups are isomorphic under some automorphism of $G$ (i.e. $\operatorname{Aut}(G)$ acts transitively on the set of maximal subgroups)? (Note: if this is strengthened to any two maximal subgroups being conjugate, then by this answer it becomes the same as having a unique maximal subgroup.)
 A: No idea about a classification, but here are some non-abelian examples. First recall that in a finite $p$-group, the maximal proper subgroups are just the kernels of homomorphisms onto $\mathbf{Z}/p\mathbf{Z}$.
(1) the free group of rank $k$ in the variety $V_{p,k,\ell}$ of $k$-step nilpotent groups satisfying the law $x^{p^\ell}=1$. In this case the automorphism group acts transitively on the set of maximal proper subgroups.
(2) the Heisenberg group of order $p^{2n+1}$ (at least for odd $p$). It can be described as the set of square matrices of size $n+2$ of the form $\begin{pmatrix} I_1 & x & z \\ 0 & I_n & y \\ 0 & 0 & I_1\end{pmatrix}$ over $K=\mathbf{Z}/p\mathbf{Z}$. In this case also, the automorphism group acts transitively on the set of maximal proper subgroups. I haven't checked $p=2$, and also I haven't checked if it works with fields $\mathbf{F}_{p^k}$ for $k\ge 2$.
There are many variants of the examples in (1) (using other varieties, e.g., exponent $p^2$ with derived subgroup of exponent $p$ etc.) Among groups in (2) there are groups of exponent $p$ and order $p^3$ and $p^5$, and among those in (1) there are groups of exponent $p$ and order $p^5$ and $p^6$.
I don't know if there are non-abelian groups of order $p^4$ answering the question but this should be checkable. Also I don't know examples for which the automorphism group doesn't act transitively on maximal subgroups. (Edit: there are such examples for both, see the comments.)
