  A Mumford group is a discontinuous subgroup $\Gamma$ of PGL(2,K), where K
denotes a non archimedean valued field, such that the quotient by $\Gamma$ is a
curve of genus 0. As abstract group $\Gamma$ is an amalgam of a finite tree of
finite groups. For K of positive characteristic the large collection of
amalgams having two or three branch points is classified. Using these data
Mumford curves with a large group of automorphisms are discovered. A long
combinatorial proof, involving the classification of the finite simple groups,
is needed for establishing an upper bound for the order of the group of
automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are
introduced. For the projective line the relations with Mumford groups and
singular stratified bundles are studied. This paper is a sequel to our paper
"Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part
of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and
K.~Kontogeorgis.
