The Nottingham group at 2 is the group of (formal) power series t+a_2 t(squared)+ \dots in the variable t with coefficients a_i from the field with two elements, where the group operation is given by composition of power series. Only a handful of such power series of finite order are explicitly known through a formula for their coefficients (following work by Klopsch, Chinburg-Symonds and Jean). We switch perspective and describe such series through an automaton, a kind of memory-less Turing machine described by a finite directed labelled graph. Combining some Galois theory with lucky guessing and and algorithmic version of a theorem of Christol (or a more recent version of Bridy and Speyer), we are able to provide an explicit automaton-theoretic description of (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) a series of order 8 with minimal break sequence; and (c) an embedding of the Klein Four Group into the Nottingham group at 2. We relate the existence of "closed formulas" to sparseness properties of the series and of the related automata, related to a recent field-theoretic characterisation of sparseness due to Albayrak and Bell. (Joint work with Jakub Byszewski and Djurre Tijsma.)