TITLE: Finite groups with few conjugacy classes of maximal subgroups
QUESTION [11 upvotes]: Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$? 
Same question, but this time $G$ is a finite group with at most $c$ conjugacy classes of core-free maximal subgroup.

Notes:

Question 1 is equivalent to asking about groups with at most $c$ primitive permutation representations (up to permutation equivalence).
Question 2 is equivalent to asking about groups with at most $c$ faithful primitive permutation representations (up to permutation equivalence).
I'm interested in the same idea, but instead of permutation equivalence, one considers permutation isomorphism. This would change the original questions so that one considers $\textrm{Aut}(G)$-conjugacy classes, instead of just $G$-conjugacy classes.

What might an answer look like:

If $c=1$, then one can prove that $G$ must be cyclic of prime power order. Core-free will require $G$ is of prime order. Can one give a full classification for $c=2,3,4,\dots$? Non-solvable groups enter at $c=3$ (e.g. $A_5$) -- I'm presuming that $c=2$ will still imply solvability, although I haven't written down a proof.
If one considers the variant mentioned in the third bullet point of the notes above -- referring to $\textrm{Aut}(G)$-conjugacy classes -- then one also obtains elementary-abelian groups for $c=1$.
I'm presuming that this stuff has been studied before so this question is also a reference-request. There is a conjecture about upper bounds for the number of maximal subgroups -- see my answer here... But my interest in the current question is specifically about very small values of $c$, so that conjecture is not so relevant...

REPLY [5 votes]: Here are a few more references:

S. Adnan, On groups having exactly $2$ conjugacy classes of maximal subgroups, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 66 (1979), no. 3, 175–178. link
S. Adnan, On groups having exactly $2$ conjugacy classes of maximal subgroups. II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68 (1980), no. 3, 179. link
W. Shi, Finite groups having at most two classes of maximal subgroups of the same order, Chinese Ann. Math. Ser. A 10 (1989), no. 5, 532–537.
V. A. Belonogov, Finite groups with three classes of maximal subgroups, Math. Sb. 131 (1968), 225–239. link
X. Li, Finite groups having three classes of maximal subgroups of the same order, Acta Math. Sinica 1 (1994), 108–115.
G. Pazderski, Über maximale Untergruppen endlicher Gruppen, Math. Nachr. 26 (1963/1964), 307–319. 
J. Shi, W. Shi, and C. Zhang, The type of conjugacy classes of maximal subgroups and characterization of finite groups, Comm. Algebra 38 (2010), no. 1, 143–153. link
J. Wang, The number of maximal subgroups and their types, Pure Appl. Math. (Xi'an) 5 (1989), 24–33. 

The main theorem of Belonogov's article states that
Theorem. A finite nonsolvable group has three conjugacy classes of maximal subgroups if and only if $G/\Phi(G)$ is isomorphic to $PSL(2,7)$ or $PSL(2,2^p)$ for some prime $p$.