Let $G$ be a group of polynomial growth, let $S$ be a finite generating set for $G$ and let $B_n$ be the set of elements of $G$ given by words of length $\leq n$ in the generating set $S$. Is there an upper bound for the numbers $\displaystyle\frac{\#B_{2n}}{\#B_n}$, with $n\in\mathbb{N}$?
Context: I am looking for sufficients conditions for a group to have a certain kind of Folner sequence, and if this is true then polynomial growth groups would have such Folner sequences. In case anyone is curious, the Folner sequences I am looking for are left-Folner sequences $(F_N)_{N\in\mathbb{N}}$ such that $\exists\varepsilon>0$ $\forall N\in\mathbb{N}$ there exist infinitely many values of $L\in\mathbb{N}$ such that there is a set $A\subseteq F_L$ which is a disjoint union of right translates of $F_N$ and satisfies $\frac{\#A}{\#F_L}>\varepsilon$. I call such a Folner sequence "self-covering".
Definition of polynomial growth I am using: A finitely generated group has polynomial growth if there exists a generating set $S$ and some big constant $N\in\mathbb{N}$ such that for all $n\in\mathbb{N}$, the ball $B_n$ defined above has $\leq N\cdot n^N$ elements.
According to Gromov's polynomial growth theorem, each finitely generated group of polynomial growth is virtually nilpotent. According to the Bass–Guivarc’h Theorem (see references in the same link) the for virtually nilpotent groups the growth function is $\sim n^d$ for some $d$ that can be computed in terms of the lower central series of the finite index nilpotent subgroup, meaning that $$ a n^d \le |B_n|\le b n^d $$ for all $n$ and some fixed positive constants $a, b$. From this, it is immediate that $$ \sup_n \frac{|B_{2n}|}{|B_n|} <\infty. $$