Let $A, B, C$ be infinite groups and suppose that there are injective homomorphisms $\iota_A \colon C \to A$ and $\iota_B \colon C \to B$ such that $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. Define a following chain of subgroups of finite index in $C$.
\begin{align*} K_0 &= C,\\ K_{i+1} &= \iota_A^{-1}(\operatorname{core}_A(\iota_A(K_i)))\cap\iota_A^{-1}(\operatorname{core}_B(\iota_B(K_i))), \end{align*} where $\operatorname{core}_G(H)$ denotes the normal core of $H$ in $G$ for $H\leq G$, and set $K = \bigcap_{0 \leq n} K_n$. One can check that $K$ is the maximal subgroup (with respect to inclusion) of $C$ such that $\iota_A(K) \unlhd A$ and $\iota_B(K) \unlhd B$.
My question is this: is it possible that $|A:\iota_B(K)| = \infty = |B:\iota_B(K)|$? If so, can one give a "nice" description of the family of such triples of groups? Does the answer change if we add additional assumption that $A,B,C$ are finitely generated or profinite?
Take $T$ to be the $3$-valent tree with (full) automorphism group $X$ and $\{u,v\}$ an edge. Let $A=X_v$, $B=X_{u,v}$ and $C=X_{(u,v)}$. We have $|A:C|=3$ and $|B:C|=2$ and $K=1$. ($A$ and $B$ generate $X$, and $C$ has trivial core in $X$.)