Splitting groups over subgroups

Question

Let $G$ be a finitely presented group with a subgroup $H$(if it helps we can assume that $H$ is finitely presented as well.) Is there any method in order to check that whether $G$ splits over the subgroup $H$ or not?

Answer 1

There is an unsolved conjecture by Kropholler and Roller concerning how to check a slightly weaker condition, namely whether $G$ splits over a subgroup that is commensurable to $H$. Since that conjecture is still unresolved, your stronger question is also unresolved.

Answer 2

At least, there's no algorithm, say when $G$ is input by a presentation and when $H$ is input by giving generators.

Indeed, if there were such an algorithm, then we would deduce an algorithm to determine whether $G$ splits over $\{1\}$. Applying this algorithm to the free product $G\ast G$ (inputting a finite presentation of $G$), we deduce an algorithm solving the triviality problem (says yes iff $G\neq\{1\}$). It is classical that there is no such algorithm.