$G$ is $p$-supersoluble iff $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$. Let $G$ be a $p$-soluble group. Then $G$ is $p$-supersoluble if and only if
$|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.
 A: This is the proof from Satz VI.9.3 in Huppert's Endliche Gruppen page 717. [time to leave, but you can just read the proof there...]
Proposition: Let $G$ be a $p$-soluble group all of whose proper quotients are $p$-supersoluble and all of whose maximal subgroups are index $p$ or relatively prime to $p$. Then $G$ is $p$-supersoluble.
Proof: Assume by way of contradiction that $G$ is not $p$-supersoluble. Let $N$ be a minimal normal subgroup of $G$. Since $G/N$ is $p$-supersoluble, but $G$ is not, we must have that $N$ has order divisible by $p$ but not order exactly $p$. Since $G$ is $p$-soluble, $N$ has order a power of $p$. If $N$ is not contained $\Phi(G)$, then there is a maximal subgroup $M$ of $G$ that does not contain $N$. Since $M \cap N$ is normal in $M$ (both $M$ and $N$ are normalized by $M$), and normal in $N$ ($N$ is abelian, so normalizes all of its subgroups), $M \cap N$ is normal in $G=MN$. Since $N$ is minimal normal and $M \cap N < N$, we must have $M \cap N = 1$ so that $p=[G:M]=[N:1]$. Since $N$ does not have order $p$, $N$ must be contained within the Frattini subgroup of $G$. Since $G/\Phi(G)$ is a proper quotient of $G$, $G/\Phi(G)$ is $p$-supersoluble. Huppert showed that $G$ is $p$-supersoluble iff $G/\Phi(G)$ is $p$-supersoluble, so this final contradiction shows $G$ must be $p$-supersoluble. $\square$
Lemma: [Huppert (1966)] If $G/N$ is $p$-supersoluble and $N \leq \Phi(G)$, then $G$ is $p$-supersoluble.
Proof: Since $N$ is nilpotent and minimal normal, $N$ is elementary abelian. If $N$ has order relatively prime to $p$, then $G$ is $p$-supersoluble by definition. Hence we may assume $N$ has order a power of $p$ and we seek to prove this power is 1. ... XXX: Class time.
An earlier proof of the global result is in Huppert (1954).


*

*Huppert, Bertram.
“Normalteiler und maximale Untergruppen endlicher Gruppen.”
Math. Z. 60, (1954). 409–434.
MR64771
DOI:10.1007/BF01187387

*Huppert, Bertram.
“Zur Gaschützschen Theorie der Formationen.”
Math. Ann. 164 (1966) 133–141.
MR199264
DOI:10.1007/BF01429051
