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$-module, we recover the graded version of Nakayama's lemma, something similar should hold for the filtered version. Surely someone has taken up this point of view (Which I learned from Gruenberg's "Cohomological topics in group theory") to define a Frattini object for a large class of categories? 1 Let me describe a common generalization of Nakayama's lemma and Burnside's basis theorem which may shed some light here (it may include the other cases, but I haven't checked). Let$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)$is$J(R)$, the Jacobdon radical of$R$. Let's denote the smallest$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$. We have the following theorem: The set of non-generators is precisely$\mathbb{Fr}_P(G)$. By taking$P$empty we obtain Burnsides basis theorem. By taking$G$to be an$R$-module and$P$to be the multiplicative semigroup of$R\$ we recover Nakayama's lemma. 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?