Is a sufficient condition that $H$, $K$ $\leq G$ such that $|H|=|K|$ and $[G:H]$ are finites to conclude that $[G:K]=[G:H]$?

Question

Let $G$ an infinite group, $H$ and $K$ proper subgroups of $G$ such that $|H|$ and $|K|$ are finite and $|H|=|K|$, also $[G:H]$ is finite. The question is: With this hypothesis can I conclude that $[G:K]=[G:H]$?

I try to figure it out that the property of finitude in the index and the equality of the cardinality of the subgroups is sufficient to conclude that the index of G in K is also finite and equal to [G:H].

In the question Are there sufficient conditions on ,⪇, ≠, such that [:]=[:]for infinite ? its clearly the answer to finite groups but I'm stuck with infinite groups.

I would be very thankful for an answer to this question.

Answer 1

Recall:

$$|G|=[G:H]|H|.$$

If both factors of the RHS are finite, then so is the LHS.