  In this paper, we investigate properties of automorphism groups of function
fields in one variable in relation to its reductions with respect to special
valuations. In 1969, Deligne and Mumford proved that there exists a natural
injective homomorphism between the automorphism groups of $\mathcal{X}_{\eta}$
and any special fibre of $\mathcal{X}.$ Here, we give a generalisation of this
theorem in function field setting of Deuring's theory of constant reductions.
