# Index of a common normal core

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?