Let $G$ be finitely generated, and consider a chain of subgroups $G > K \geq H$. We say that $G$ splits over $H$ if $G$ acts without inversions on a simplicial tree $T$ with some edge stabilised by $H$, and this action is minimal (no proper invariant sub tree).
Suppose that $G$ does indeed split over $H$. Can we conclude that $G$ splits over $K$?
I feel like I've seen before that this is true, for relatively trivial reasons. However, I'm struggling to reproduce this and now I'm starting to doubt it's truth.
There's a chance I may be missing some important hypotheses, but any comments are appreciated. Thanks
Quick edit: Upon reflection, a possible necessary condition on $K$ could be that $K$ is a co-dimension 1 subgroup of $G$.
Edit 2: Added the condition that this action be minimal.