This question is follow up of [this MO-post][1].

First let us recall the necessary definitions.

A function $f:X\to X$ on a group $X$ is called a *polynomial* if there exists $n\in\mathbb N$ and elements $a_0,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$.

Let $\mathrm{Poly(X)}$ be the set of all polynomials on a group $X$. It is a submonoid of the monoid $X^X$ of all self-maps of $X$.

Observe that each polynomial $f$ on a commutative group $X$ is of the form $f(x)=ax^n$ for some $a\in X$ and $n\in \mathbb N$, which implies that $|\mathrm{Poly}(X)|=|X|\cdot\exp(X)$ for each finite commutative group $X$. 

Here $\exp(X):=\min\{n\ge 1:\forall x\in X\;\;(x^n=1)\}$ is the exponent of the group $X$.

Since any group $X$ acts effectively on $\mathrm{Poly}(X)$ by left (or right) shifts, the cardinal $|\mathrm{Poly}(X)|$ is divisible by $|X|$.

>**Problem 1.** Is $|\mathrm{Poly}(X)|$ divisible by $|X|\cdot\exp(X)$ for every finite group $X$?

The cardinality of the monoid $\mathrm{Poly}(X)$ was calculated by [Peter Taylor][2] for all non-commutative groups $X$ of cardinality $|X|<24$. 

>**Problem 2.** Is $|\mathrm{Poly}(X)|$ divisible by $|X|^2$ for every non-commutative finite group $X$?

Peter Taylow observed that the answer to Problem 2 is negative for the group $D_{12}=GAP(12,4)$ with $|\mathrm{Poly}(D_{12})|=648=2^3\times 3^4$. So, only Problems 1 and 3 remain open.

The calculations of Peter Taylor show that the following problem has affirmative answer for all finite groups of cardinality $<24$:

>**Problem 3.** Let $X$ be a finite group and $p$ be a prime number dividing $|\mathrm{Poly}(X)|$. Is $p$ a divisor of $|X|$?

**Remark 1.** The answer to Problems 2 and 3 are affirmative for finite simple groups since $\mathrm{Poly}(X)=X^X$ for any non-commutative simple finite group $X$, see the [answer of @YCor][3] to [this MO-question][1]. 


  [1]: https://mathoverflow.net/q/429619/61536
  [2]: https://gist.github.com/pjt33/b9e1e62291110b164e091ef7c2f7ead5
  [3]: https://mathoverflow.net/a/429635/61536