A finite group $G$ is called integral if there is a finite group $H$ such that $G\cong H'$. In Araujo, Cameron, Casolo, Matucci's paper, integrals of groups, they tried to solve a problem as following: Given a finite group $G$, can we determine whether $G$ is integral?
There are many similar problems. For example, given a finite group $G$, determine whether there is a finite group $H$ such that $G\cong \Phi(H)$ (or $G\cong H/Z(H)$, etc.). Araujo et al. call those problem "inverse problem in finite group theory".
My question is, for a certain inverse problem, whether we can prove or disprove the existence of an algorithm which outputs the answer to the problem in finite time.
For example, can we prove or disprove there is an algorithm which can output "The group $G$ is integral" in finite time if we input an integral group $G$ and output "The group $G$ is non-integral" in finite time if we input a non-integral group $G$? I'm particularly interested in the problem of determining whether a finite group is integral. Can we know the existence of an algorithm which can answer the integral group problem? I know we might need to construct an algorithm to prove the existence of algorithm if there is an algorithm for the integral group problem. But how to prove the non-existence of algorithm if there is no such an algorithm?
In Eick's paper, she gave a positive answer to the problem of determining whether there is a group $H$ such that $G\cong \Phi(H)$. She proved that $G\cong \Phi(H)$ for some $H$ if and only if $Inn(G) ≤ \Phi(Aut(G))$.
Some idea: Define a characteristic functor $F$ a functor sending a finite group $G$ to a characteristic subgroup of $G$ such that for any isomorphism $\sigma:G\rightarrow H$, $F(G^\sigma)=(F(G))^\sigma$. Now, we define a characteristic functor as following: $$\chi(G)=\left\{\begin{aligned}&G,\mbox{ if $G$ is integral}\\ &1,\mbox{ otherwise}\end{aligned}\right.$$ Then it can be seen the problem is decidable if and only if $\chi$ is computable hence we might solve this problem if we know enough properties of computable characteristic functor.
For example, given two characteristic functors $F_1$ and $F_2$, their composition $F_1F_2(G):=F_1(F_2(G))$ is also a characteristic functor. Moreover, if both $F_1$ and $F_2$ are computable, $F_1F_2$ is also computable. If we can find some `initial' characteristic functors and operations on characteristic functors which preserving computability and prove they can generate all computable characteristic functors, we then derive a good description about computable characteristic functors.
