Does there exists a group $G$ satisfying all the following conditions?


1. $G$ is finitely generated,
2. $G$ is of bounded torsion (has finite exponent),
3. $G$ has finitely many elements of order $2$,
4. $G$ has infinitely many elements of order $2^l$ for some $l$ (for example $l=2$).


By 1. and 2. $G$ is a quotient of a free Burnside group.


Near-examples of such groups violating only Condition 2. are [generalized dicyclic group](https://en.wikipedia.org/wiki/Dicyclic_group#Generalizations) $Dic(A,y)$, where $A$ is a finitely generated abelian group. 


References on somehow related problems are welcome.



Torsion group with finitely many elements of order 2 but infinitely many elements of order 4
Here is an example with infinitely many $4$-torsion elements and only one $2$-torsion element. (The smallest number of generators I can do is $4$, and the exponent is pretty big and not a power of $2$.)


**Example.** Pick $n$ odd and $d \geq 2$ such that $B(d,n)$ is infinite. Let $p$ be an odd prime. Set
$$A = \mathbf Z/2 \oplus \bigoplus\_{g \in B(d,n)} (\mathbf Z/p)e\_g,$$
with distinguished element $y = (1,0,\ldots)$ or order $2$. Then the generalised dicyclic group $\operatorname{Dic}(A,y) = A \amalg Ax$ satisfies all criteria except 1:


1. $\operatorname{Dic}(A,y)$ is not finitely generated because it has an index $2$ subgroup $A$ that is not finitely generated.
2. Every element of $\operatorname{Dic}(A,y)$ is killed by $4p$.
3. The only element of order $2$ is $y \in A$.
4. Every element in $Ax$ has order $4$, and $Ax$ is infinite by assumption.


Finally, the group $B(d,n)$ acts on $A$ fixing $y$ by $(g,e\_h) \mapsto e\_{gh}$. Thus this extends to an action on $\operatorname{Dic}(A,y)$, and we take $G$ to be the semidirect product
$$G = \operatorname{Dic}(A,y) \rtimes B(d,n).$$
Then all criteria are satisfied:


1. If $x\_1,\ldots,x\_d$ are the standard generators of $B(d,n)$, then $x, e\_1, x\_1,\ldots,x\_d$ generate $G$. Indeed, they generate the quotient group $B(d,n)$, hence we get all elements $ge\_1g^{-1} = e\_g$, hence we get everything.
2. Since $B(d,n)$ has exponent $n$ and $\operatorname{Dic}(A,y)$ has exponent $4p$, we conclude that $G$ has exponent dividing $4pn$.
3. and 4. Since $B(d,n)$ has odd exponent, all $2$-power torsion happens in $\operatorname{Dic}(A,y)$.


So $G$ is an example. $\square$


**Remark.** I have no idea if there are also examples of $2$-power exponent.


