  The enumeration of walks in the quarter plane confined in the first quadrant
has attracted a lot of attention over the past fifteenth years. The generating
functions associated to small steps models satisfy a functional equation in two
catalytic variables. For such models, Bousquet-M\'elou and Mishna defined a
group called the group of the walk which turned out to be central in the
classification of small steps models. In particular, its action on the
catalytic variables yields a set of change of variables compatible with the
structure of the functional equation. This particular set called the orbit has
been generalized to models with arbitrary large steps by Bostan,
Bousquet-M\'elou and Melczer. However, the orbit had till now no underlying
group.
  In this article, we endow the orbit with the action of a Galois group, which
extends the group of the walk to models with large steps. Within this Galoisian
framework, we generalized the notions of invariants and decoupling. This enable
us to develop a general strategy to prove the algebraicity of models with small
backward steps. Our constructions lead to the first proofs of algebraicity of
weighted models with large steps, proving in particular a conjecture of Bostan,
Bousquet-M\'elou and Melczer, and allowing us to find new algebraic models with
large steps.
