Let
$(G, \mu )$ be a discrete group equipped with a generating probability measure, and let
$\Gamma $ be a finite index subgroup of
$G$. A
$\mu $-random walk on
$G$, starting from the identity, returns to
$\Gamma $ with probability one. Let
$\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits
$\Gamma $. We prove that the Furstenberg entropy of a
$(G, \mu )$-stationary space, with respect to the action of
$(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of
$(G, \mu )$, times the index of
$\Gamma $ in
$G$. The index is shown to be equal to the expected return time to
$\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of
$(G, \mu )$, we prove that the random walk entropy of
$(\Gamma , \theta )$ is equal to the random walk entropy of
$(G, \mu )$, times the index of
$\Gamma $ in
$G$.