If $k\subset \ell\subset m$ are finite fields, then $\mathrm{Aut}(\ell /k)$ is the quotient $\mathrm{Aut}(m/k)/\mathrm{Aut}(m/\ell)$ (correct me if I'm wrong).
Does this work in a more general setting? As in, what if the restriction of $k, \ell, m$ to finite fields is removed?