  A class of additive cellular automata (ACA) on a finite group is defined by
an index-group $\m g$ and a finite field $\m F_p$ for a prime modulus $p$
\cite{Bul_arch_1}. This paper deals mainly with ACA on infinite commutative
groups and direct products of them with some non commutative $p$-groups. It
appears that for all abelian groups, the rules and initial states with finite
supports define behaviors which being restricted to some infinite regular
series of time moments become significantly simplified. In particular, for free
abelian groups with $n$ generators states $V^{[t]}$ of ACA with a rule $R$ at
time moments $t=p^k,k>k_0,$ can be viewed as $||R||$ copies of initial state
$V^{[0]}$ moving through an $n$-dimensional Euclidean space. That is the
behavior is similar to gliders from J.Conway's automaton {\sl Life}. For some
other special infinite series of time moments the automata states approximate
self-similar structures and the approximation becomes better with time. An
infinite class $\mathrm{DHC}(\mbf S,\theta)$ of non-commutative $p$-groups is
described which in particular includes quaternion and dihedral $p$-groups. It
is shown that the simplification of behaviors takes place as well for direct
products of non-commutative groups from the class $\mathrm{DHC}(\mbf S,\theta)$
with commutative groups. Finally, an automaton on a non-commutative group is
constructed such that its behavior at time moments $2^k,k\ge2,$ is similar to a
glider gun. It is concluded that ACA on non-commutative groups demonstrate more
diverse variety of behaviors comparing to ACA on commutative groups.
