  This paper concerns the general problem of classifying the finite
deterministic automata that admit a synchronizing (or reset) word. (For our
purposes it is irrelevant if the automata has initial or final states.) Our
departure point is the study of the transition semigroup associated to the
automaton, taking advantage of the enormous and very deep progresses made
during the last decades on the theory of permutation groups, their geometry and
their combinatorial structure.
  Let $X$ be a finite set. We say that a primitive group $G$ on $X$ is {\em
synchronizing} if $G$ together with any non-invertible map on $X$ generates a
constant map. It is known (by some recent results proved by P. M. Neumann) that
for some primitive groups $G$ and for some singular transformations $t$ of
uniform kernel (that is, all blocks have the same number of elements), the
semigroup $< G,t>$ does not generate a constant map. Therefore the following
concept is very natural: a primitive group $G$ on $X$ is said to be {\em almost
synchronizing} if $G$ together with any map of non-uniform kernel generates a
constant map. In this paper we use two different methods to provide several
infinite families of groups that are not synchronizing, but are almost
synchronizing. The paper ends with a number of problems on synchronization
likely to attract the attention of experts in computer science, combinatorics
and geometry, groups and semigroups, linear algebra and matrix theory.
