Let me quote some well-known results and perhaps related problems which may be illuminating!

Let $G$ be a non-abelian finite $p$-group having cyclic center. Then, there is no finite $p$-group $H$ such that $G$ is isomorphic to a normal subgroup of the derived subgroup $[H,H]$ of $H$. In particular, $G$ cannot be isomorphic to the derived subgoup of some $p$-group $H$. The latter is a famous result due to Burnside. The former is a slight generalization of problem. See H. Heineken, On normal embedding of subgroups.
Geom. Dedicata 83, No.1-3, 211-216 (2000).

Related problem: Let $V$ be a non-empty set of words in the free group on the countable set
$\{x_1,x_2,\dots\}$. We call a group $G$ is {\bf integrable with respect to $V$}, whenever there is a group $H$ such that $G\cong V(H)$, where $V(H)$ is the verbal subgroup of $H$ generated by $V$, i.e., $$V(H)=\langle v(h_1,\dots,h_n) | v\in V, h_i \in H \rangle,$$
(the subgroup of $H$ generated by the values of words of $V$ on the elements of $H$)
For example, if one takes $V=\{[x_1,x_2]=x_1^{-1}x_2^{-1}x_1 x_2\}$, then $V(H)$ is the derived subgroup of $H$ for any group $H$ and the problem is the same as it proposed.
One may write (maybe for some propaganda)
$$\int G \; dV=H \Longleftrightarrow G=V(H).$$
In the case, $V=\{ [x_1,x_2]\}$, $$\int G=H \Longleftrightarrow G=H'$$ and so
$$\int G=H \Longleftrightarrow \int G=H \times A$$ for any abelian group $A$. (This may remind the constant term in the integral of a function!)
I have used the latter notation which has no benefit but only perhaps inspiring in
[A. Abdollahi, Integral of a group, 29th Iranian International Conference on Mathematics, Amirkabir University of Technology (Tehran Polytechnic), Iran, March 28-31, 1998.]

I would like to say that the problem `given a group $G$, find groups $H$ such that $G=[H,H]$" have been studied in a more general contex which may be found with key words`

normal embedding of subgroups" and the above-mentioned paper of Heineken is a good start.

Also it may be worth-mentioning that by a result of Allenby
R.B.J.T. Allenby, Normal subgroups contained in Frattini subgroups are Frattini subgroups, Proc. Amer. Math. Soc, Vol. 78, No. 3, 1980, 318-

if $N$ is a normal subgroup of a finite group $G$ which is contained in the Frattini subgroup of $G$, then $N=\Phi(U)$ for some finite group $U$.

Of course, for the class of finite $p$-groups the Frattini subgroup is the verbal subgroup
generated by the words $x_1^p, [x_1,x_2]$.