Extreme points in quotients of Hardy spaces

In the Hardy spaces $H^1$ and $H^{\mathrm{\infty }}$, there are neat and well-known characterizations of the extreme points of the unit ball. We obtain counterparts of these classical theorems when $H^1$ (resp., $H^{\mathrm{\infty }}$) gets replaced by the quotient space $H^1\mathrm{/}E$ (resp., $H^{\mathrm{\infty }}\mathrm{/}E$), under certain assumptions on the subspace $E$. In the $H^1$ setting, we also treat the case where the underlying space is taken to be the kernel of a Toeplitz operator.