1. Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$?
2. 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...