Suppose $A$ is a finite-index subgroup of $G$, with a finite generating set $S$. Take a transversal $K$ of $A$ (say, containing the identity), and take $T := S \cup K$ as a finite generating set for $G$. This induces word metrics $d_S$ and $d_T$ on $A$ and $G$ respectively.
My question is, do $d_S$ and $d_T$ agree on $A$? In particular, does $d_S = d_T |_A$? I'm struggling to find literature on the topic, nor an easy counterexample.
One direction is obvious: for any $a,b \in A$ it is clear that $d_T(a,b) \leq d_S(a,b)$.
Any direction on this would be appreciated - either a counter example or steps towards a proof.
Thanks!