  We study in detail the profinite group G arising as geometric \'etale
iterated monodromy group of an arbitrary quadratic morphism f with an infinite
postcritical orbit over a field of characteristic different from two. This is a
self-similar closed subgroup of the group of automorphisms of a regular rooted
binary tree. In many cases it is equal to the automorphism group of the tree,
but there remain some interesting cases where it is not. In these cases we
prove that the conjugacy class of G depends only on the combinatorial type of
the postcritical orbit of f. We also determine the Hausdorff dimension and the
normalizer of G. This result is then used to describe the arithmetic \'etale
iterated monodromy group of f.
  The methods used mostly group theoretical and of the same type as in a
previous article of the same author dealing with quadratic polynomials with a
finite postcritical orbit. The results on abstract self-similar profinite
groups acting on a regular rooted binary tree may be of independent interest.
