Let me describe a common generalization of Nakayama's lemma lemmas and Burnside's basis theorem which may shed some light here(it may include the other cases, but I haven't checked). Let $G$ be a group and $P$ a set of endomorphisms of $G$. A $P$-subgroup will be a subgroup of $G$ which is closed under acting by elements of $P$. We'll call $\mathbb{Fr}_P(G)$ the "$P$-Frattini subgroup of $G$", defined as the intersection of all maximal $P$-subgroups. Note the following special cases:
- When $P$ is empty then $\mathbb{Fr}_P(G)$ is the Frattini subgroup of $G$.
- When $R$ is a ring, $G$ its additive group and $P$ its multiplicative semigroup then $\mathbb{Fr}(G)$ \mathbb{Fr}_P(G)$ is $J(R)$, the Jacobdon radical of $R$.
- When $P$ is as above, and $G$ is an $R$-module then $\mathbb{Fr}_P(G)$ contains $J(R)G$.
Let's denote the smallest $P$-subgroup containing a set $S$ by $\langle S\rangle_P$, and call an element $x\in G$, a non-generator if $G=\langle S,x \rangle_P$ always implies $G=\langle S\rangle_P$. We have the following theorem:
The set of non-generators of $G$ is precisely $\mathbb{Fr}_P(G)$.
By taking $P$ empty we obtain Burnsides Burnside's basis theorem. By taking $G$ to be an $R$-module and $P$ to be the multiplicative semigroup of $R$ we recover Nakayama's lemma. If $R$ is a graded ring and we take $P$ to be the semigroup of elements of positive degree and $G$ to be a graded $R$-module, we recover the graded version of Nakayama's lemma, something similar should hold for the filtered version. Surely someone has taken up this point of view (Which I learned from Gruenberg's "Cohomological topics in group theory") to define a Frattini object for a large class of categories?
