According to Burnside's classification in his book "Theory of groups of finite order", one of the types of non-abelian groups of order $pq^2$ ($p$ and $q$ are distinct primes), has the presentation $\langle a, b \vert a^p=b^{q^2}=1, aba^{-1}=b^i, {\rm Ord}_{q^2}(i)=p\rangle$.
Is it true that all maximal subgroups of such group are cyclic?
Is it possible to classify all finite non-abelian groups $G$ of order $pq^m$ with all cyclic Sylow subgroups, $m\geq 2$ and $p<q$ such that all maximal subgroups of $G$ are cyclic?
(We know that such groups are supersolvable, because all Sylow subgroups of them are cyclic. Thus all maximal subgroups of such groups are of prime index).
Thank you very much!